Practice 8-6 Natural Logarithmsstaff.bbhcsd.org/bradacm/files/2014/04/8.6-algebra-2.pdfApr 08, 2014...
Transcript of Practice 8-6 Natural Logarithmsstaff.bbhcsd.org/bradacm/files/2014/04/8.6-algebra-2.pdfApr 08, 2014...
![Page 1: Practice 8-6 Natural Logarithmsstaff.bbhcsd.org/bradacm/files/2014/04/8.6-algebra-2.pdfApr 08, 2014 · Changing log e x y= to exponential form would give e xy =. Evaluating log e](https://reader035.fdocuments.in/reader035/viewer/2022070911/5fb18f72f20cc62f47153453/html5/thumbnails/1.jpg)
Name _____________________________ Class ___________________Date ____________
Practice 8-6 Natural Logarithms Remember that common logarithms are logarithms of base 10. 4 4
10log3 log 3x x+ +=
e is the base of the Natural Logarithms, often abbreviated as ln. ( ) ( )log ln xe x =
Often called Euler’s number, e is an irrational that has a value of 2.718281828459045…
Changing loge x y= to exponential form would give ye x= .
Evaluating loge x There are two keys on the TI-84 that are for evaluating exponential functions with base e. They are the second function of the / (e) and L (ex) keys. Evaluate each expression to the nearest thousandth.
1. 5e ________________ 2. 4e− ________________ 3. 1
3e ________________ The same properties (product, quotient and power) of exponents apply to natural logarithms. So 33ln ln lnx y x y+ = Write as a single natural logarithm.
4. 4ln 2 ln f− = ________________ 5. 1
lnx 3ln2
y+ = ________________
Solving natural logarithmic equations.
Solve ( )2ln 3 5 4x + =
Write in exponential form. ( )24 3 5e x= +
Take the square root of both sides. 4 3 5e x± = +
Subtract 5 from both sides. 4 5 3e x± − =
Divide both sides by 3. 4 5
3
ex
± − =
Evaluate using the calculator. 7.39 and 4.130x x= − =
6. Solve ln 9 5x = 7. Solve 2
ln 123
x + =
![Page 2: Practice 8-6 Natural Logarithmsstaff.bbhcsd.org/bradacm/files/2014/04/8.6-algebra-2.pdfApr 08, 2014 · Changing log e x y= to exponential form would give e xy =. Evaluating log e](https://reader035.fdocuments.in/reader035/viewer/2022070911/5fb18f72f20cc62f47153453/html5/thumbnails/2.jpg)
Solving an exponential equation. Solve 27 2.5 20xe + = Subtract 2.5 from both sides to start 27 17.5xe = isolating the exponential term. Divide both sides by 7 to finish 2 2.5xe = isolating the exponential term. Take the natural log of both sides. 2ln ln 2.5xe = Use the power property. 2 x ln ln 2.5e = Simplify using ln 1e = . 2 x ln 2.5=
Divide by 2. ln 2.5
x2
=
Use the calculator to finish. x 0.458≈ 8. Solve 1 30xe + = 9. Solve 34 1.2 14xe + =
Solve each equation. Round to the nearest ten-thousandth. Check your answer. Show all work for each. Circle your answer. 10. 18xe = 11. 2 12xe = 12. ln 3 6x =
13. ( )ln 4 1 36x − =
14. 5 4 7x
e + = 15. 22ln 2 x 1=
lne = 1
![Page 3: Practice 8-6 Natural Logarithmsstaff.bbhcsd.org/bradacm/files/2014/04/8.6-algebra-2.pdfApr 08, 2014 · Changing log e x y= to exponential form would give e xy =. Evaluating log e](https://reader035.fdocuments.in/reader035/viewer/2022070911/5fb18f72f20cc62f47153453/html5/thumbnails/3.jpg)
Name _____________________________ Class ___________________Date ____________
Practice 8-6 Natural Logarithms Remember that common logarithms are logarithms of base 10. 4 4
10log3 log 3x x+ +=
e is the base of the Natural Logarithms, often abbreviated as ln. ( ) ( )log ln xe x =
Often called Euler’s number, e is an irrational that has a value of 2.718281828459045…
Changing loge x y= to exponential form would give ye x= .
Evaluating loge x There are two keys on the TI-84 that are for evaluating exponential functions with base e. They are the second function of the / (e) and L (ex) keys. Evaluate each expression to the nearest thousandth.
1. 5e ________________ 2. 4e− ________________ 3. 1
3e ________________ The same properties (product, quotient and power) of exponents apply to natural logarithms. So 33ln ln lnx y x y+ = Write as a single natural logarithm.
4. 4 ln 2 ln f− = ________________ 5. 1
lnx 3ln2
y+ = ________________
Solving natural logarithmic equations.
Solve ( )2ln 3 5 4x + =
Write in exponential form. ( )24 3 5e x= +
Take the square root of both sides. 4 3 5e x± = +
Subtract 5 from both sides. 4 5 3e x± − =
Divide both sides by 3. 4 5
3
ex
± − =
Evaluate using the calculator. 7.39 and 4.130x x= − =
6. Solve ln 9 5x = 7. Solve 2
ln 123
x + =
![Page 4: Practice 8-6 Natural Logarithmsstaff.bbhcsd.org/bradacm/files/2014/04/8.6-algebra-2.pdfApr 08, 2014 · Changing log e x y= to exponential form would give e xy =. Evaluating log e](https://reader035.fdocuments.in/reader035/viewer/2022070911/5fb18f72f20cc62f47153453/html5/thumbnails/4.jpg)
Solving an exponential equation. Solve 27 2.5 20xe + = Subtract 2.5 from both sides to start 27 17.5xe = isolating the exponential term. Divide both sides by 7 to finish 2 2.5xe = isolating the exponential term. Take the natural log of both sides. 2ln ln 2.5xe = Use the power property. 2 x ln ln 2.5e = Simplify using ln 1e = . 2 x ln 2.5=
Divide by 2. ln 2.5
x2
=
Use the calculator to finish. x 0.458≈ 8. Solve 1 30xe + = 9. Solve 34 1.2 14xe + =
Solve each equation. Round to the nearest ten-thousandth. Check your answer. Show all work for each. Circle your answer. 10. 18xe = 11. 2 12xe = 12. ln 3 6x =
13. ( )ln 4 1 36x − =
14. 5 4 7x
e + = 15. 22ln 2 x 1=
lne = 1
![Page 5: Practice 8-6 Natural Logarithmsstaff.bbhcsd.org/bradacm/files/2014/04/8.6-algebra-2.pdfApr 08, 2014 · Changing log e x y= to exponential form would give e xy =. Evaluating log e](https://reader035.fdocuments.in/reader035/viewer/2022070911/5fb18f72f20cc62f47153453/html5/thumbnails/5.jpg)
Name _____________________________ Class ___________________Date ____________
Practice 8-6 Natural Logarithms Remember that common logarithms are logarithms of base 10. 4 4
10log3 log 3x x+ +=
e is the base of the Natural Logarithms, often abbreviated as ln. ( ) ( )log ln xe x =
Often called Euler’s number, e is an irrational that has a value of 2.718281828459045…
Changing loge x y= to exponential form would give ye x= .
Evaluating loge x There are two keys on the TI-84 that are for evaluating exponential functions with base e. They are the second function of the / (e) and L (ex) keys. Evaluate each expression to the nearest thousandth.
1. 5e ________________ 2. 4e− ________________ 3. 1
3e ________________ The same properties (product, quotient and power) of exponents apply to natural logarithms. So 33ln ln lnx y x y+ = Write as a single natural logarithm.
4. 4 ln 2 ln f− = ________________ 5. 1
lnx 3ln2
y+ = ________________
Solving natural logarithmic equations.
Solve ( )2ln 3 5 4x + =
Write in exponential form. ( )24 3 5e x= +
Take the square root of both sides. 4 3 5e x± = +
Subtract 5 from both sides. 4 5 3e x± − =
Divide both sides by 3. 4 5
3
ex
± − =
Evaluate using the calculator. 7.39 and 4.130x x= − =
6. Solve ln 9 5x = 7. Solve 2
ln 123
x + =
![Page 6: Practice 8-6 Natural Logarithmsstaff.bbhcsd.org/bradacm/files/2014/04/8.6-algebra-2.pdfApr 08, 2014 · Changing log e x y= to exponential form would give e xy =. Evaluating log e](https://reader035.fdocuments.in/reader035/viewer/2022070911/5fb18f72f20cc62f47153453/html5/thumbnails/6.jpg)
Solving an exponential equation. Solve 27 2.5 20xe + = Subtract 2.5 from both sides to start 27 17.5xe = isolating the exponential term. Divide both sides by 7 to finish 2 2.5xe = isolating the exponential term. Take the natural log of both sides. 2ln ln 2.5xe = Use the power property. 2 x ln ln 2.5e = Simplify using ln 1e = . 2 x ln 2.5=
Divide by 2. ln 2.5
x2
=
Use the calculator to finish. x 0.458≈ 8. Solve 1 30xe + = 9. Solve 34 1.2 14xe + =
Solve each equation. Round to the nearest ten-thousandth. Check your answer. Show all work for each. Circle your answer. 10. 18xe = 11. 2 12xe = 12. ln 3 6x =
13. ( )ln 4 1 36x − =
14. 5 4 7x
e + = 15. 22ln 2 x 1=
lne = 1
![Page 7: Practice 8-6 Natural Logarithmsstaff.bbhcsd.org/bradacm/files/2014/04/8.6-algebra-2.pdfApr 08, 2014 · Changing log e x y= to exponential form would give e xy =. Evaluating log e](https://reader035.fdocuments.in/reader035/viewer/2022070911/5fb18f72f20cc62f47153453/html5/thumbnails/7.jpg)
Name _____________________________ Class ___________________Date ____________
Practice 8-6 Natural Logarithms Remember that common logarithms are logarithms of base 10. 4 4
10log3 log 3x x+ +=
e is the base of the Natural Logarithms, often abbreviated as ln. ( ) ( )log ln xe x =
Often called Euler’s number, e is an irrational that has a value of 2.718281828459045…
Changing loge x y= to exponential form would give ye x= .
Evaluating loge x There are two keys on the TI-84 that are for evaluating exponential functions with base e. They are the second function of the / (e) and L (ex) keys. Evaluate each expression to the nearest thousandth.
1. 5e ________________ 2. 4e− ________________ 3. 1
3e ________________ The same properties (product, quotient and power) of exponents apply to natural logarithms. So 33ln ln lnx y x y+ = Write as a single natural logarithm.
4. 4 ln 2 ln f− = ________________ 5. 1
lnx 3ln2
y+ = ________________
Solving natural logarithmic equations.
Solve ( )2ln 3 5 4x + =
Write in exponential form. ( )24 3 5e x= +
Take the square root of both sides. 4 3 5e x± = +
Subtract 5 from both sides. 4 5 3e x± − =
Divide both sides by 3. 4 5
3
ex
± − =
Evaluate using the calculator. 7.39 and 4.130x x= − =
6. Solve ln 9 5x = 7. Solve 2
ln 123
x + =
![Page 8: Practice 8-6 Natural Logarithmsstaff.bbhcsd.org/bradacm/files/2014/04/8.6-algebra-2.pdfApr 08, 2014 · Changing log e x y= to exponential form would give e xy =. Evaluating log e](https://reader035.fdocuments.in/reader035/viewer/2022070911/5fb18f72f20cc62f47153453/html5/thumbnails/8.jpg)
Solving an exponential equation. Solve 27 2.5 20xe + = Subtract 2.5 from both sides to start 27 17.5xe = isolating the exponential term. Divide both sides by 7 to finish 2 2.5xe = isolating the exponential term. Take the natural log of both sides. 2ln ln 2.5xe = Use the power property. 2 x ln ln 2.5e = Simplify using ln 1e = . 2 x ln 2.5=
Divide by 2. ln 2.5
x2
=
Use the calculator to finish. x 0.458≈ 8. Solve 1 30xe + = 9. Solve 34 1.2 14xe + =
Solve each equation. Round to the nearest ten-thousandth. Check your answer. Show all work for each. Circle your answer. 10. 18xe = 11. 2 12xe = 12. ln 3 6x =
13. ( )ln 4 1 36x − =
14. 5 4 7x
e + = 15. 22ln 2 x 1=
lne = 1
![Page 9: Practice 8-6 Natural Logarithmsstaff.bbhcsd.org/bradacm/files/2014/04/8.6-algebra-2.pdfApr 08, 2014 · Changing log e x y= to exponential form would give e xy =. Evaluating log e](https://reader035.fdocuments.in/reader035/viewer/2022070911/5fb18f72f20cc62f47153453/html5/thumbnails/9.jpg)
Name _____________________________ Class ___________________Date ____________
Practice 8-6 Natural Logarithms Remember that common logarithms are logarithms of base 10. 4 4
10log3 log 3x x+ +=
e is the base of the Natural Logarithms, often abbreviated as ln. ( ) ( )log ln xe x =
Often called Euler’s number, e is an irrational that has a value of 2.718281828459045…
Changing loge x y= to exponential form would give ye x= .
Evaluating loge x There are two keys on the TI-84 that are for evaluating exponential functions with base e. They are the second function of the / (e) and L (ex) keys. Evaluate each expression to the nearest thousandth.
1. 5e ________________ 2. 4e− ________________ 3. 1
3e ________________ The same properties (product, quotient and power) of exponents apply to natural logarithms. So 33ln ln lnx y x y+ = Write as a single natural logarithm.
4. 4 ln 2 ln f− = ________________ 5. 1
lnx 3ln2
y+ = ________________
Solving natural logarithmic equations.
Solve ( )2ln 3 5 4x + =
Write in exponential form. ( )24 3 5e x= +
Take the square root of both sides. 4 3 5e x± = +
Subtract 5 from both sides. 4 5 3e x± − =
Divide both sides by 3. 4 5
3
ex
± − =
Evaluate using the calculator. 7.39 and 4.130x x= − =
6. Solve ln 9 5x = 7. Solve 2
ln 123
x + =
![Page 10: Practice 8-6 Natural Logarithmsstaff.bbhcsd.org/bradacm/files/2014/04/8.6-algebra-2.pdfApr 08, 2014 · Changing log e x y= to exponential form would give e xy =. Evaluating log e](https://reader035.fdocuments.in/reader035/viewer/2022070911/5fb18f72f20cc62f47153453/html5/thumbnails/10.jpg)
Solving an exponential equation. Solve 27 2.5 20xe + = Subtract 2.5 from both sides to start 27 17.5xe = isolating the exponential term. Divide both sides by 7 to finish 2 2.5xe = isolating the exponential term. Take the natural log of both sides. 2ln ln 2.5xe = Use the power property. 2 x ln ln 2.5e = Simplify using ln 1e = . 2 x ln 2.5=
Divide by 2. ln 2.5
x2
=
Use the calculator to finish. x 0.458≈ 8. Solve 1 30xe + = 9. Solve 34 1.2 14xe + =
Solve each equation. Round to the nearest ten-thousandth. Check your answer. Show all work for each. Circle your answer. 10. 18xe = 11. 2 12xe = 12. ln 3 6x =
13. ( )ln 4 1 36x − =
14. 5 4 7x
e + = 15. 22ln 2 x 1=
lne = 1
![Page 11: Practice 8-6 Natural Logarithmsstaff.bbhcsd.org/bradacm/files/2014/04/8.6-algebra-2.pdfApr 08, 2014 · Changing log e x y= to exponential form would give e xy =. Evaluating log e](https://reader035.fdocuments.in/reader035/viewer/2022070911/5fb18f72f20cc62f47153453/html5/thumbnails/11.jpg)
Name _____________________________ Class ___________________Date ____________
Practice 8-6 Natural Logarithms Remember that common logarithms are logarithms of base 10. 4 4
10log3 log 3x x+ +=
e is the base of the Natural Logarithms, often abbreviated as ln. ( ) ( )log ln xe x =
Often called Euler’s number, e is an irrational that has a value of 2.718281828459045…
Changing loge x y= to exponential form would give ye x= .
Evaluating loge x There are two keys on the TI-84 that are for evaluating exponential functions with base e. They are the second function of the / (e) and L (ex) keys. Evaluate each expression to the nearest thousandth.
1. 5e ________________ 2. 4e− ________________ 3. 1
3e ________________ The same properties (product, quotient and power) of exponents apply to natural logarithms. So 33ln ln lnx y x y+ = Write as a single natural logarithm.
4. 4 ln 2 ln f− = ________________ 5. 1
lnx 3ln2
y+ = ________________
Solving natural logarithmic equations.
Solve ( )2ln 3 5 4x + =
Write in exponential form. ( )24 3 5e x= +
Take the square root of both sides. 4 3 5e x± = +
Subtract 5 from both sides. 4 5 3e x± − =
Divide both sides by 3. 4 5
3
ex
± − =
Evaluate using the calculator. 7.39 and 4.130x x= − =
6. Solve ln 9 5x = 7. Solve 2
ln 123
x + =
![Page 12: Practice 8-6 Natural Logarithmsstaff.bbhcsd.org/bradacm/files/2014/04/8.6-algebra-2.pdfApr 08, 2014 · Changing log e x y= to exponential form would give e xy =. Evaluating log e](https://reader035.fdocuments.in/reader035/viewer/2022070911/5fb18f72f20cc62f47153453/html5/thumbnails/12.jpg)
Solving an exponential equation. Solve 27 2.5 20xe + = Subtract 2.5 from both sides to start 27 17.5xe = isolating the exponential term. Divide both sides by 7 to finish 2 2.5xe = isolating the exponential term. Take the natural log of both sides. 2ln ln 2.5xe = Use the power property. 2 x ln ln 2.5e = Simplify using ln 1e = . 2 x ln 2.5=
Divide by 2. ln 2.5
x2
=
Use the calculator to finish. x 0.458≈ 8. Solve 1 30xe + = 9. Solve 34 1.2 14xe + =
Solve each equation. Round to the nearest ten-thousandth. Check your answer. Show all work for each. Circle your answer. 10. 18xe = 11. 2 12xe = 12. ln 3 6x =
13. ( )ln 4 1 36x − =
14. 5 4 7x
e + = 15. 22ln 2 x 1=
lne = 1
![Page 13: Practice 8-6 Natural Logarithmsstaff.bbhcsd.org/bradacm/files/2014/04/8.6-algebra-2.pdfApr 08, 2014 · Changing log e x y= to exponential form would give e xy =. Evaluating log e](https://reader035.fdocuments.in/reader035/viewer/2022070911/5fb18f72f20cc62f47153453/html5/thumbnails/13.jpg)
Name _____________________________ Class ___________________Date ____________
Practice 8-6 Natural Logarithms Remember that common logarithms are logarithms of base 10. 4 4
10log3 log 3x x+ +=
e is the base of the Natural Logarithms, often abbreviated as ln. ( ) ( )log ln xe x =
Often called Euler’s number, e is an irrational that has a value of 2.718281828459045…
Changing loge x y= to exponential form would give ye x= .
Evaluating loge x There are two keys on the TI-84 that are for evaluating exponential functions with base e. They are the second function of the / (e) and L (ex) keys. Evaluate each expression to the nearest thousandth.
1. 5e ________________ 2. 4e− ________________ 3. 1
3e ________________ The same properties (product, quotient and power) of exponents apply to natural logarithms. So 33ln ln lnx y x y+ = Write as a single natural logarithm.
4. 4 ln 2 ln f− = ________________ 5. 1
lnx 3ln2
y+ = ________________
Solving natural logarithmic equations.
Solve ( )2ln 3 5 4x + =
Write in exponential form. ( )24 3 5e x= +
Take the square root of both sides. 4 3 5e x± = +
Subtract 5 from both sides. 4 5 3e x± − =
Divide both sides by 3. 4 5
3
ex
± − =
Evaluate using the calculator. 7.39 and 4.130x x= − =
6. Solve ln 9 5x = 7. Solve 2
ln 123
x + =
![Page 14: Practice 8-6 Natural Logarithmsstaff.bbhcsd.org/bradacm/files/2014/04/8.6-algebra-2.pdfApr 08, 2014 · Changing log e x y= to exponential form would give e xy =. Evaluating log e](https://reader035.fdocuments.in/reader035/viewer/2022070911/5fb18f72f20cc62f47153453/html5/thumbnails/14.jpg)
Solving an exponential equation. Solve 27 2.5 20xe + = Subtract 2.5 from both sides to start 27 17.5xe = isolating the exponential term. Divide both sides by 7 to finish 2 2.5xe = isolating the exponential term. Take the natural log of both sides. 2ln ln 2.5xe = Use the power property. 2 x ln ln 2.5e = Simplify using ln 1e = . 2 x ln 2.5=
Divide by 2. ln 2.5
x2
=
Use the calculator to finish. x 0.458≈ 8. Solve 1 30xe + = 9. Solve 34 1.2 14xe + =
Solve each equation. Round to the nearest ten-thousandth. Check your answer. Show all work for each. Circle your answer. 10. 18xe = 11. 2 12xe = 12. ln 3 6x =
13. ( )ln 4 1 36x − =
14. 5 4 7x
e + = 15. 22ln 2 x 1=
lne = 1
![Page 15: Practice 8-6 Natural Logarithmsstaff.bbhcsd.org/bradacm/files/2014/04/8.6-algebra-2.pdfApr 08, 2014 · Changing log e x y= to exponential form would give e xy =. Evaluating log e](https://reader035.fdocuments.in/reader035/viewer/2022070911/5fb18f72f20cc62f47153453/html5/thumbnails/15.jpg)
Name _____________________________ Class ___________________Date ____________
Practice 8-6 Natural Logarithms Remember that common logarithms are logarithms of base 10. 4 4
10log3 log 3x x+ +=
e is the base of the Natural Logarithms, often abbreviated as ln. ( ) ( )log ln xe x =
Often called Euler’s number, e is an irrational that has a value of 2.718281828459045…
Changing loge x y= to exponential form would give ye x= .
Evaluating loge x There are two keys on the TI-84 that are for evaluating exponential functions with base e. They are the second function of the / (e) and L (ex) keys. Evaluate each expression to the nearest thousandth.
1. 5e ________________ 2. 4e− ________________ 3. 1
3e ________________ The same properties (product, quotient and power) of exponents apply to natural logarithms. So 33ln ln lnx y x y+ = Write as a single natural logarithm.
4. 4 ln 2 ln f− = ________________ 5. 1
lnx 3ln2
y+ = ________________
Solving natural logarithmic equations.
Solve ( )2ln 3 5 4x + =
Write in exponential form. ( )24 3 5e x= +
Take the square root of both sides. 4 3 5e x± = +
Subtract 5 from both sides. 4 5 3e x± − =
Divide both sides by 3. 4 5
3
ex
± − =
Evaluate using the calculator. 7.39 and 4.130x x= − =
6. Solve ln 9 5x = 7. Solve 2
ln 123
x + =
![Page 16: Practice 8-6 Natural Logarithmsstaff.bbhcsd.org/bradacm/files/2014/04/8.6-algebra-2.pdfApr 08, 2014 · Changing log e x y= to exponential form would give e xy =. Evaluating log e](https://reader035.fdocuments.in/reader035/viewer/2022070911/5fb18f72f20cc62f47153453/html5/thumbnails/16.jpg)
Solving an exponential equation. Solve 27 2.5 20xe + = Subtract 2.5 from both sides to start 27 17.5xe = isolating the exponential term. Divide both sides by 7 to finish 2 2.5xe = isolating the exponential term. Take the natural log of both sides. 2ln ln 2.5xe = Use the power property. 2 x ln ln 2.5e = Simplify using ln 1e = . 2 x ln 2.5=
Divide by 2. ln 2.5
x2
=
Use the calculator to finish. x 0.458≈ 8. Solve 1 30xe + = 9. Solve 34 1.2 14xe + =
Solve each equation. Round to the nearest ten-thousandth. Check your answer. Show all work for each. Circle your answer. 10. 18xe = 11. 2 12xe = 12. ln 3 6x =
13. ( )ln 4 1 36x − =
14. 5 4 7x
e + = 15. 22ln 2 x 1=
lne = 1
![Page 17: Practice 8-6 Natural Logarithmsstaff.bbhcsd.org/bradacm/files/2014/04/8.6-algebra-2.pdfApr 08, 2014 · Changing log e x y= to exponential form would give e xy =. Evaluating log e](https://reader035.fdocuments.in/reader035/viewer/2022070911/5fb18f72f20cc62f47153453/html5/thumbnails/17.jpg)
Name _____________________________ Class ___________________Date ____________
Practice 8-6 Natural Logarithms Remember that common logarithms are logarithms of base 10. 4 4
10log3 log 3x x+ +=
e is the base of the Natural Logarithms, often abbreviated as ln. ( ) ( )log ln xe x =
Often called Euler’s number, e is an irrational that has a value of 2.718281828459045…
Changing loge x y= to exponential form would give ye x= .
Evaluating loge x There are two keys on the TI-84 that are for evaluating exponential functions with base e. They are the second function of the / (e) and L (ex) keys. Evaluate each expression to the nearest thousandth.
1. 5e ________________ 2. 4e− ________________ 3. 1
3e ________________ The same properties (product, quotient and power) of exponents apply to natural logarithms. So 33ln ln lnx y x y+ = Write as a single natural logarithm.
4. 4 ln 2 ln f− = ________________ 5. 1
lnx 3ln2
y+ = ________________
Solving natural logarithmic equations.
Solve ( )2ln 3 5 4x + =
Write in exponential form. ( )24 3 5e x= +
Take the square root of both sides. 4 3 5e x± = +
Subtract 5 from both sides. 4 5 3e x± − =
Divide both sides by 3. 4 5
3
ex
± − =
Evaluate using the calculator. 7.39 and 4.130x x= − =
6. Solve ln 9 5x = 7. Solve 2
ln 123
x + =
![Page 18: Practice 8-6 Natural Logarithmsstaff.bbhcsd.org/bradacm/files/2014/04/8.6-algebra-2.pdfApr 08, 2014 · Changing log e x y= to exponential form would give e xy =. Evaluating log e](https://reader035.fdocuments.in/reader035/viewer/2022070911/5fb18f72f20cc62f47153453/html5/thumbnails/18.jpg)
Solving an exponential equation. Solve 27 2.5 20xe + = Subtract 2.5 from both sides to start 27 17.5xe = isolating the exponential term. Divide both sides by 7 to finish 2 2.5xe = isolating the exponential term. Take the natural log of both sides. 2ln ln 2.5xe = Use the power property. 2 x ln ln 2.5e = Simplify using ln 1e = . 2 x ln 2.5=
Divide by 2. ln 2.5
x2
=
Use the calculator to finish. x 0.458≈ 8. Solve 1 30xe + = 9. Solve 34 1.2 14xe + =
Solve each equation. Round to the nearest ten-thousandth. Check your answer. Show all work for each. Circle your answer. 10. 18xe = 11. 2 12xe = 12. ln 3 6x =
13. ( )ln 4 1 36x − =
14. 5 4 7x
e + = 15. 22ln 2 x 1=
lne = 1
![Page 19: Practice 8-6 Natural Logarithmsstaff.bbhcsd.org/bradacm/files/2014/04/8.6-algebra-2.pdfApr 08, 2014 · Changing log e x y= to exponential form would give e xy =. Evaluating log e](https://reader035.fdocuments.in/reader035/viewer/2022070911/5fb18f72f20cc62f47153453/html5/thumbnails/19.jpg)
Name _____________________________ Class ___________________Date ____________
Practice 8-6 Natural Logarithms Remember that common logarithms are logarithms of base 10. 4 4
10log3 log 3x x+ +=
e is the base of the Natural Logarithms, often abbreviated as ln. ( ) ( )log ln xe x =
Often called Euler’s number, e is an irrational that has a value of 2.718281828459045…
Changing loge x y= to exponential form would give ye x= .
Evaluating loge x There are two keys on the TI-84 that are for evaluating exponential functions with base e. They are the second function of the / (e) and L (ex) keys. Evaluate each expression to the nearest thousandth.
1. 5e ________________ 2. 4e− ________________ 3. 1
3e ________________ The same properties (product, quotient and power) of exponents apply to natural logarithms. So 33ln ln lnx y x y+ = Write as a single natural logarithm.
4. 4 ln 2 ln f− = ________________ 5. 1
lnx 3ln2
y+ = ________________
Solving natural logarithmic equations.
Solve ( )2ln 3 5 4x + =
Write in exponential form. ( )24 3 5e x= +
Take the square root of both sides. 4 3 5e x± = +
Subtract 5 from both sides. 4 5 3e x± − =
Divide both sides by 3. 4 5
3
ex
± − =
Evaluate using the calculator. 7.39 and 4.130x x= − =
6. Solve ln 9 5x = 7. Solve 2
ln 123
x + =
![Page 20: Practice 8-6 Natural Logarithmsstaff.bbhcsd.org/bradacm/files/2014/04/8.6-algebra-2.pdfApr 08, 2014 · Changing log e x y= to exponential form would give e xy =. Evaluating log e](https://reader035.fdocuments.in/reader035/viewer/2022070911/5fb18f72f20cc62f47153453/html5/thumbnails/20.jpg)
Solving an exponential equation. Solve 27 2.5 20xe + = Subtract 2.5 from both sides to start 27 17.5xe = isolating the exponential term. Divide both sides by 7 to finish 2 2.5xe = isolating the exponential term. Take the natural log of both sides. 2ln ln 2.5xe = Use the power property. 2 x ln ln 2.5e = Simplify using ln 1e = . 2 x ln 2.5=
Divide by 2. ln 2.5
x2
=
Use the calculator to finish. x 0.458≈ 8. Solve 1 30xe + = 9. Solve 34 1.2 14xe + =
Solve each equation. Round to the nearest ten-thousandth. Check your answer. Show all work for each. Circle your answer. 10. 18xe = 11. 2 12xe = 12. ln 3 6x =
13. ( )ln 4 1 36x − =
14. 5 4 7x
e + = 15. 22ln 2 x 1=
lne = 1
![Page 21: Practice 8-6 Natural Logarithmsstaff.bbhcsd.org/bradacm/files/2014/04/8.6-algebra-2.pdfApr 08, 2014 · Changing log e x y= to exponential form would give e xy =. Evaluating log e](https://reader035.fdocuments.in/reader035/viewer/2022070911/5fb18f72f20cc62f47153453/html5/thumbnails/21.jpg)
Name _____________________________ Class ___________________Date ____________
Practice 8-6 Natural Logarithms Remember that common logarithms are logarithms of base 10. 4 4
10log3 log 3x x+ +=
e is the base of the Natural Logarithms, often abbreviated as ln. ( ) ( )log ln xe x =
Often called Euler’s number, e is an irrational that has a value of 2.718281828459045…
Changing loge x y= to exponential form would give ye x= .
Evaluating loge x There are two keys on the TI-84 that are for evaluating exponential functions with base e. They are the second function of the / (e) and L (ex) keys. Evaluate each expression to the nearest thousandth.
1. 5e ________________ 2. 4e− ________________ 3. 1
3e ________________ The same properties (product, quotient and power) of exponents apply to natural logarithms. So 33ln ln lnx y x y+ = Write as a single natural logarithm.
4. 4 ln 2 ln f− = ________________ 5. 1
lnx 3ln2
y+ = ________________
Solving natural logarithmic equations.
Solve ( )2ln 3 5 4x + =
Write in exponential form. ( )24 3 5e x= +
Take the square root of both sides. 4 3 5e x± = +
Subtract 5 from both sides. 4 5 3e x± − =
Divide both sides by 3. 4 5
3
ex
± − =
Evaluate using the calculator. 7.39 and 4.130x x= − =
6. Solve ln 9 5x = 7. Solve 2
ln 123
x + =
![Page 22: Practice 8-6 Natural Logarithmsstaff.bbhcsd.org/bradacm/files/2014/04/8.6-algebra-2.pdfApr 08, 2014 · Changing log e x y= to exponential form would give e xy =. Evaluating log e](https://reader035.fdocuments.in/reader035/viewer/2022070911/5fb18f72f20cc62f47153453/html5/thumbnails/22.jpg)
Solving an exponential equation. Solve 27 2.5 20xe + = Subtract 2.5 from both sides to start 27 17.5xe = isolating the exponential term. Divide both sides by 7 to finish 2 2.5xe = isolating the exponential term. Take the natural log of both sides. 2ln ln 2.5xe = Use the power property. 2 x ln ln 2.5e = Simplify using ln 1e = . 2 x ln 2.5=
Divide by 2. ln 2.5
x2
=
Use the calculator to finish. x 0.458≈ 8. Solve 1 30xe + = 9. Solve 34 1.2 14xe + =
Solve each equation. Round to the nearest ten-thousandth. Check your answer. Show all work for each. Circle your answer. 10. 18xe = 11. 2 12xe = 12. ln 3 6x =
13. ( )ln 4 1 36x − =
14. 5 4 7x
e + = 15. 22ln 2 x 1=
lne = 1
![Page 23: Practice 8-6 Natural Logarithmsstaff.bbhcsd.org/bradacm/files/2014/04/8.6-algebra-2.pdfApr 08, 2014 · Changing log e x y= to exponential form would give e xy =. Evaluating log e](https://reader035.fdocuments.in/reader035/viewer/2022070911/5fb18f72f20cc62f47153453/html5/thumbnails/23.jpg)
Name _____________________________ Class ___________________Date ____________
Practice 8-6 Natural Logarithms Remember that common logarithms are logarithms of base 10. 4 4
10log3 log 3x x+ +=
e is the base of the Natural Logarithms, often abbreviated as ln. ( ) ( )log ln xe x =
Often called Euler’s number, e is an irrational that has a value of 2.718281828459045…
Changing loge x y= to exponential form would give ye x= .
Evaluating loge x There are two keys on the TI-84 that are for evaluating exponential functions with base e. They are the second function of the / (e) and L (ex) keys. Evaluate each expression to the nearest thousandth.
1. 5e ________________ 2. 4e− ________________ 3. 1
3e ________________ The same properties (product, quotient and power) of exponents apply to natural logarithms. So 33ln ln lnx y x y+ = Write as a single natural logarithm.
4. 4 ln 2 ln f− = ________________ 5. 1
lnx 3ln2
y+ = ________________
Solving natural logarithmic equations.
Solve ( )2ln 3 5 4x + =
Write in exponential form. ( )24 3 5e x= +
Take the square root of both sides. 4 3 5e x± = +
Subtract 5 from both sides. 4 5 3e x± − =
Divide both sides by 3. 4 5
3
ex
± − =
Evaluate using the calculator. 7.39 and 4.130x x= − =
6. Solve ln 9 5x = 7. Solve 2
ln 123
x + =
![Page 24: Practice 8-6 Natural Logarithmsstaff.bbhcsd.org/bradacm/files/2014/04/8.6-algebra-2.pdfApr 08, 2014 · Changing log e x y= to exponential form would give e xy =. Evaluating log e](https://reader035.fdocuments.in/reader035/viewer/2022070911/5fb18f72f20cc62f47153453/html5/thumbnails/24.jpg)
Solving an exponential equation. Solve 27 2.5 20xe + = Subtract 2.5 from both sides to start 27 17.5xe = isolating the exponential term. Divide both sides by 7 to finish 2 2.5xe = isolating the exponential term. Take the natural log of both sides. 2ln ln 2.5xe = Use the power property. 2 x ln ln 2.5e = Simplify using ln 1e = . 2 x ln 2.5=
Divide by 2. ln 2.5
x2
=
Use the calculator to finish. x 0.458≈ 8. Solve 1 30xe + = 9. Solve 34 1.2 14xe + =
Solve each equation. Round to the nearest ten-thousandth. Check your answer. Show all work for each. Circle your answer. 10. 18xe = 11. 2 12xe = 12. ln 3 6x =
13. ( )ln 4 1 36x − =
14. 5 4 7x
e + = 15. 22ln 2 x 1=
lne = 1
![Page 25: Practice 8-6 Natural Logarithmsstaff.bbhcsd.org/bradacm/files/2014/04/8.6-algebra-2.pdfApr 08, 2014 · Changing log e x y= to exponential form would give e xy =. Evaluating log e](https://reader035.fdocuments.in/reader035/viewer/2022070911/5fb18f72f20cc62f47153453/html5/thumbnails/25.jpg)
Name _____________________________ Class ___________________Date ____________
Practice 8-6 Natural Logarithms Remember that common logarithms are logarithms of base 10. 4 4
10log3 log 3x x+ +=
e is the base of the Natural Logarithms, often abbreviated as ln. ( ) ( )log ln xe x =
Often called Euler’s number, e is an irrational that has a value of 2.718281828459045…
Changing loge x y= to exponential form would give ye x= .
Evaluating loge x There are two keys on the TI-84 that are for evaluating exponential functions with base e. They are the second function of the / (e) and L (ex) keys. Evaluate each expression to the nearest thousandth.
1. 5e ________________ 2. 4e− ________________ 3. 1
3e ________________ The same properties (product, quotient and power) of exponents apply to natural logarithms. So 33ln ln lnx y x y+ = Write as a single natural logarithm.
4. 4 ln 2 ln f− = ________________ 5. 1
lnx 3ln2
y+ = ________________
Solving natural logarithmic equations.
Solve ( )2ln 3 5 4x + =
Write in exponential form. ( )24 3 5e x= +
Take the square root of both sides. 4 3 5e x± = +
Subtract 5 from both sides. 4 5 3e x± − =
Divide both sides by 3. 4 5
3
ex
± − =
Evaluate using the calculator. 7.39 and 4.130x x= − =
6. Solve ln 9 5x = 7. Solve 2
ln 123
x + =
![Page 26: Practice 8-6 Natural Logarithmsstaff.bbhcsd.org/bradacm/files/2014/04/8.6-algebra-2.pdfApr 08, 2014 · Changing log e x y= to exponential form would give e xy =. Evaluating log e](https://reader035.fdocuments.in/reader035/viewer/2022070911/5fb18f72f20cc62f47153453/html5/thumbnails/26.jpg)
Solving an exponential equation. Solve 27 2.5 20xe + = Subtract 2.5 from both sides to start 27 17.5xe = isolating the exponential term. Divide both sides by 7 to finish 2 2.5xe = isolating the exponential term. Take the natural log of both sides. 2ln ln 2.5xe = Use the power property. 2 x ln ln 2.5e = Simplify using ln 1e = . 2 x ln 2.5=
Divide by 2. ln 2.5
x2
=
Use the calculator to finish. x 0.458≈ 8. Solve 1 30xe + = 9. Solve 34 1.2 14xe + =
Solve each equation. Round to the nearest ten-thousandth. Check your answer. Show all work for each. Circle your answer. 10. 18xe = 11. 2 12xe = 12. ln 3 6x =
13. ( )ln 4 1 36x − =
14. 5 4 7x
e + = 15. 22ln 2 x 1=
lne = 1
![Page 27: Practice 8-6 Natural Logarithmsstaff.bbhcsd.org/bradacm/files/2014/04/8.6-algebra-2.pdfApr 08, 2014 · Changing log e x y= to exponential form would give e xy =. Evaluating log e](https://reader035.fdocuments.in/reader035/viewer/2022070911/5fb18f72f20cc62f47153453/html5/thumbnails/27.jpg)
Name _____________________________ Class ___________________Date ____________
Practice 8-6 Natural Logarithms Remember that common logarithms are logarithms of base 10. 4 4
10log3 log 3x x+ +=
e is the base of the Natural Logarithms, often abbreviated as ln. ( ) ( )log ln xe x =
Often called Euler’s number, e is an irrational that has a value of 2.718281828459045…
Changing loge x y= to exponential form would give ye x= .
Evaluating loge x There are two keys on the TI-84 that are for evaluating exponential functions with base e. They are the second function of the / (e) and L (ex) keys. Evaluate each expression to the nearest thousandth.
1. 5e ________________ 2. 4e− ________________ 3. 1
3e ________________ The same properties (product, quotient and power) of exponents apply to natural logarithms. So 33ln ln lnx y x y+ = Write as a single natural logarithm.
4. 4 ln 2 ln f− = ________________ 5. 1
lnx 3ln2
y+ = ________________
Solving natural logarithmic equations.
Solve ( )2ln 3 5 4x + =
Write in exponential form. ( )24 3 5e x= +
Take the square root of both sides. 4 3 5e x± = +
Subtract 5 from both sides. 4 5 3e x± − =
Divide both sides by 3. 4 5
3
ex
± − =
Evaluate using the calculator. 7.39 and 4.130x x= − =
6. Solve ln 9 5x = 7. Solve 2
ln 123
x + =
![Page 28: Practice 8-6 Natural Logarithmsstaff.bbhcsd.org/bradacm/files/2014/04/8.6-algebra-2.pdfApr 08, 2014 · Changing log e x y= to exponential form would give e xy =. Evaluating log e](https://reader035.fdocuments.in/reader035/viewer/2022070911/5fb18f72f20cc62f47153453/html5/thumbnails/28.jpg)
Solving an exponential equation. Solve 27 2.5 20xe + = Subtract 2.5 from both sides to start 27 17.5xe = isolating the exponential term. Divide both sides by 7 to finish 2 2.5xe = isolating the exponential term. Take the natural log of both sides. 2ln ln 2.5xe = Use the power property. 2 x ln ln 2.5e = Simplify using ln 1e = . 2 x ln 2.5=
Divide by 2. ln 2.5
x2
=
Use the calculator to finish. x 0.458≈ 8. Solve 1 30xe + = 9. Solve 34 1.2 14xe + =
Solve each equation. Round to the nearest ten-thousandth. Check your answer. Show all work for each. Circle your answer. 10. 18xe = 11. 2 12xe = 12. ln 3 6x =
13. ( )ln 4 1 36x − =
14. 5 4 7x
e + = 15. 22ln 2 x 1=
lne = 1
![Page 29: Practice 8-6 Natural Logarithmsstaff.bbhcsd.org/bradacm/files/2014/04/8.6-algebra-2.pdfApr 08, 2014 · Changing log e x y= to exponential form would give e xy =. Evaluating log e](https://reader035.fdocuments.in/reader035/viewer/2022070911/5fb18f72f20cc62f47153453/html5/thumbnails/29.jpg)
Name _____________________________ Class ___________________Date ____________
Practice 8-6 Natural Logarithms Remember that common logarithms are logarithms of base 10. 4 4
10log3 log 3x x+ +=
e is the base of the Natural Logarithms, often abbreviated as ln. ( ) ( )log ln xe x =
Often called Euler’s number, e is an irrational that has a value of 2.718281828459045…
Changing loge x y= to exponential form would give ye x= .
Evaluating loge x There are two keys on the TI-84 that are for evaluating exponential functions with base e. They are the second function of the / (e) and L (ex) keys. Evaluate each expression to the nearest thousandth.
1. 5e ________________ 2. 4e− ________________ 3. 1
3e ________________ The same properties (product, quotient and power) of exponents apply to natural logarithms. So 33ln ln lnx y x y+ = Write as a single natural logarithm.
4. 4 ln 2 ln f− = ________________ 5. 1
lnx 3ln2
y+ = ________________
Solving natural logarithmic equations.
Solve ( )2ln 3 5 4x + =
Write in exponential form. ( )24 3 5e x= +
Take the square root of both sides. 4 3 5e x± = +
Subtract 5 from both sides. 4 5 3e x± − =
Divide both sides by 3. 4 5
3
ex
± − =
Evaluate using the calculator. 7.39 and 4.130x x= − =
6. Solve ln 9 5x = 7. Solve 2
ln 123
x + =
![Page 30: Practice 8-6 Natural Logarithmsstaff.bbhcsd.org/bradacm/files/2014/04/8.6-algebra-2.pdfApr 08, 2014 · Changing log e x y= to exponential form would give e xy =. Evaluating log e](https://reader035.fdocuments.in/reader035/viewer/2022070911/5fb18f72f20cc62f47153453/html5/thumbnails/30.jpg)
Solving an exponential equation. Solve 27 2.5 20xe + = Subtract 2.5 from both sides to start 27 17.5xe = isolating the exponential term. Divide both sides by 7 to finish 2 2.5xe = isolating the exponential term. Take the natural log of both sides. 2ln ln 2.5xe = Use the power property. 2 x ln ln 2.5e = Simplify using ln 1e = . 2 x ln 2.5=
Divide by 2. ln 2.5
x2
=
Use the calculator to finish. x 0.458≈ 8. Solve 1 30xe + = 9. Solve 34 1.2 14xe + =
Solve each equation. Round to the nearest ten-thousandth. Check your answer. Show all work for each. Circle your answer. 10. 18xe = 11. 2 12xe = 12. ln 3 6x =
13. ( )ln 4 1 36x − =
14. 5 4 7x
e + = 15. 22ln 2 x 1=
lne = 1
![Page 31: Practice 8-6 Natural Logarithmsstaff.bbhcsd.org/bradacm/files/2014/04/8.6-algebra-2.pdfApr 08, 2014 · Changing log e x y= to exponential form would give e xy =. Evaluating log e](https://reader035.fdocuments.in/reader035/viewer/2022070911/5fb18f72f20cc62f47153453/html5/thumbnails/31.jpg)
Name _____________________________ Class ___________________Date ____________
Practice 8-6 Natural Logarithms Remember that common logarithms are logarithms of base 10. 4 4
10log3 log 3x x+ +=
e is the base of the Natural Logarithms, often abbreviated as ln. ( ) ( )log ln xe x =
Often called Euler’s number, e is an irrational that has a value of 2.718281828459045…
Changing loge x y= to exponential form would give ye x= .
Evaluating loge x There are two keys on the TI-84 that are for evaluating exponential functions with base e. They are the second function of the / (e) and L (ex) keys. Evaluate each expression to the nearest thousandth.
1. 5e ________________ 2. 4e− ________________ 3. 1
3e ________________ The same properties (product, quotient and power) of exponents apply to natural logarithms. So 33ln ln lnx y x y+ = Write as a single natural logarithm.
4. 4 ln 2 ln f− = ________________ 5. 1
lnx 3ln2
y+ = ________________
Solving natural logarithmic equations.
Solve ( )2ln 3 5 4x + =
Write in exponential form. ( )24 3 5e x= +
Take the square root of both sides. 4 3 5e x± = +
Subtract 5 from both sides. 4 5 3e x± − =
Divide both sides by 3. 4 5
3
ex
± − =
Evaluate using the calculator. 7.39 and 4.130x x= − =
6. Solve ln 9 5x = 7. Solve 2
ln 123
x + =
![Page 32: Practice 8-6 Natural Logarithmsstaff.bbhcsd.org/bradacm/files/2014/04/8.6-algebra-2.pdfApr 08, 2014 · Changing log e x y= to exponential form would give e xy =. Evaluating log e](https://reader035.fdocuments.in/reader035/viewer/2022070911/5fb18f72f20cc62f47153453/html5/thumbnails/32.jpg)
Solving an exponential equation. Solve 27 2.5 20xe + = Subtract 2.5 from both sides to start 27 17.5xe = isolating the exponential term. Divide both sides by 7 to finish 2 2.5xe = isolating the exponential term. Take the natural log of both sides. 2ln ln 2.5xe = Use the power property. 2 x ln ln 2.5e = Simplify using ln 1e = . 2 x ln 2.5=
Divide by 2. ln 2.5
x2
=
Use the calculator to finish. x 0.458≈ 8. Solve 1 30xe + = 9. Solve 34 1.2 14xe + =
Solve each equation. Round to the nearest ten-thousandth. Check your answer. Show all work for each. Circle your answer. 10. 18xe = 11. 2 12xe = 12. ln 3 6x =
13. ( )ln 4 1 36x − =
14. 5 4 7x
e + = 15. 22ln 2 x 1=
lne = 1
![Page 33: Practice 8-6 Natural Logarithmsstaff.bbhcsd.org/bradacm/files/2014/04/8.6-algebra-2.pdfApr 08, 2014 · Changing log e x y= to exponential form would give e xy =. Evaluating log e](https://reader035.fdocuments.in/reader035/viewer/2022070911/5fb18f72f20cc62f47153453/html5/thumbnails/33.jpg)
Name _____________________________ Class ___________________Date ____________
Practice 8-6 Natural Logarithms Remember that common logarithms are logarithms of base 10. 4 4
10log3 log 3x x+ +=
e is the base of the Natural Logarithms, often abbreviated as ln. ( ) ( )log ln xe x =
Often called Euler’s number, e is an irrational that has a value of 2.718281828459045…
Changing loge x y= to exponential form would give ye x= .
Evaluating loge x There are two keys on the TI-84 that are for evaluating exponential functions with base e. They are the second function of the / (e) and L (ex) keys. Evaluate each expression to the nearest thousandth.
1. 5e ________________ 2. 4e− ________________ 3. 1
3e ________________ The same properties (product, quotient and power) of exponents apply to natural logarithms. So 33ln ln lnx y x y+ = Write as a single natural logarithm.
4. 4 ln 2 ln f− = ________________ 5. 1
lnx 3ln2
y+ = ________________
Solving natural logarithmic equations.
Solve ( )2ln 3 5 4x + =
Write in exponential form. ( )24 3 5e x= +
Take the square root of both sides. 4 3 5e x± = +
Subtract 5 from both sides. 4 5 3e x± − =
Divide both sides by 3. 4 5
3
ex
± − =
Evaluate using the calculator. 7.39 and 4.130x x= − =
6. Solve ln 9 5x = 7. Solve 2
ln 123
x + =
![Page 34: Practice 8-6 Natural Logarithmsstaff.bbhcsd.org/bradacm/files/2014/04/8.6-algebra-2.pdfApr 08, 2014 · Changing log e x y= to exponential form would give e xy =. Evaluating log e](https://reader035.fdocuments.in/reader035/viewer/2022070911/5fb18f72f20cc62f47153453/html5/thumbnails/34.jpg)
Solving an exponential equation. Solve 27 2.5 20xe + = Subtract 2.5 from both sides to start 27 17.5xe = isolating the exponential term. Divide both sides by 7 to finish 2 2.5xe = isolating the exponential term. Take the natural log of both sides. 2ln ln 2.5xe = Use the power property. 2 x ln ln 2.5e = Simplify using ln 1e = . 2 x ln 2.5=
Divide by 2. ln 2.5
x2
=
Use the calculator to finish. x 0.458≈ 8. Solve 1 30xe + = 9. Solve 34 1.2 14xe + =
Solve each equation. Round to the nearest ten-thousandth. Check your answer. Show all work for each. Circle your answer. 10. 18xe = 11. 2 12xe = 12. ln 3 6x =
13. ( )ln 4 1 36x − =
14. 5 4 7x
e + = 15. 22ln 2 x 1=
lne = 1
![Page 35: Practice 8-6 Natural Logarithmsstaff.bbhcsd.org/bradacm/files/2014/04/8.6-algebra-2.pdfApr 08, 2014 · Changing log e x y= to exponential form would give e xy =. Evaluating log e](https://reader035.fdocuments.in/reader035/viewer/2022070911/5fb18f72f20cc62f47153453/html5/thumbnails/35.jpg)
Name _____________________________ Class ___________________Date ____________
Practice 8-6 Natural Logarithms Remember that common logarithms are logarithms of base 10. 4 4
10log3 log 3x x+ +=
e is the base of the Natural Logarithms, often abbreviated as ln. ( ) ( )log ln xe x =
Often called Euler’s number, e is an irrational that has a value of 2.718281828459045…
Changing loge x y= to exponential form would give ye x= .
Evaluating loge x There are two keys on the TI-84 that are for evaluating exponential functions with base e. They are the second function of the / (e) and L (ex) keys. Evaluate each expression to the nearest thousandth.
1. 5e ________________ 2. 4e− ________________ 3. 1
3e ________________ The same properties (product, quotient and power) of exponents apply to natural logarithms. So 33ln ln lnx y x y+ = Write as a single natural logarithm.
4. 4 ln 2 ln f− = ________________ 5. 1
lnx 3ln2
y+ = ________________
Solving natural logarithmic equations.
Solve ( )2ln 3 5 4x + =
Write in exponential form. ( )24 3 5e x= +
Take the square root of both sides. 4 3 5e x± = +
Subtract 5 from both sides. 4 5 3e x± − =
Divide both sides by 3. 4 5
3
ex
± − =
Evaluate using the calculator. 7.39 and 4.130x x= − =
6. Solve ln 9 5x = 7. Solve 2
ln 123
x + =
![Page 36: Practice 8-6 Natural Logarithmsstaff.bbhcsd.org/bradacm/files/2014/04/8.6-algebra-2.pdfApr 08, 2014 · Changing log e x y= to exponential form would give e xy =. Evaluating log e](https://reader035.fdocuments.in/reader035/viewer/2022070911/5fb18f72f20cc62f47153453/html5/thumbnails/36.jpg)
Solving an exponential equation. Solve 27 2.5 20xe + = Subtract 2.5 from both sides to start 27 17.5xe = isolating the exponential term. Divide both sides by 7 to finish 2 2.5xe = isolating the exponential term. Take the natural log of both sides. 2ln ln 2.5xe = Use the power property. 2 x ln ln 2.5e = Simplify using ln 1e = . 2 x ln 2.5=
Divide by 2. ln 2.5
x2
=
Use the calculator to finish. x 0.458≈ 8. Solve 1 30xe + = 9. Solve 34 1.2 14xe + =
Solve each equation. Round to the nearest ten-thousandth. Check your answer. Show all work for each. Circle your answer. 10. 18xe = 11. 2 12xe = 12. ln 3 6x =
13. ( )ln 4 1 36x − =
14. 5 4 7x
e + = 15. 22ln 2 x 1=
lne = 1
![Page 37: Practice 8-6 Natural Logarithmsstaff.bbhcsd.org/bradacm/files/2014/04/8.6-algebra-2.pdfApr 08, 2014 · Changing log e x y= to exponential form would give e xy =. Evaluating log e](https://reader035.fdocuments.in/reader035/viewer/2022070911/5fb18f72f20cc62f47153453/html5/thumbnails/37.jpg)
Name _____________________________ Class ___________________Date ____________
Practice 8-6 Natural Logarithms Remember that common logarithms are logarithms of base 10. 4 4
10log3 log 3x x+ +=
e is the base of the Natural Logarithms, often abbreviated as ln. ( ) ( )log ln xe x =
Often called Euler’s number, e is an irrational that has a value of 2.718281828459045…
Changing loge x y= to exponential form would give ye x= .
Evaluating loge x There are two keys on the TI-84 that are for evaluating exponential functions with base e. They are the second function of the / (e) and L (ex) keys. Evaluate each expression to the nearest thousandth.
1. 5e ________________ 2. 4e− ________________ 3. 1
3e ________________ The same properties (product, quotient and power) of exponents apply to natural logarithms. So 33ln ln lnx y x y+ = Write as a single natural logarithm.
4. 4 ln 2 ln f− = ________________ 5. 1
lnx 3ln2
y+ = ________________
Solving natural logarithmic equations.
Solve ( )2ln 3 5 4x + =
Write in exponential form. ( )24 3 5e x= +
Take the square root of both sides. 4 3 5e x± = +
Subtract 5 from both sides. 4 5 3e x± − =
Divide both sides by 3. 4 5
3
ex
± − =
Evaluate using the calculator. 7.39 and 4.130x x= − =
6. Solve ln 9 5x = 7. Solve 2
ln 123
x + =
![Page 38: Practice 8-6 Natural Logarithmsstaff.bbhcsd.org/bradacm/files/2014/04/8.6-algebra-2.pdfApr 08, 2014 · Changing log e x y= to exponential form would give e xy =. Evaluating log e](https://reader035.fdocuments.in/reader035/viewer/2022070911/5fb18f72f20cc62f47153453/html5/thumbnails/38.jpg)
Solving an exponential equation. Solve 27 2.5 20xe + = Subtract 2.5 from both sides to start 27 17.5xe = isolating the exponential term. Divide both sides by 7 to finish 2 2.5xe = isolating the exponential term. Take the natural log of both sides. 2ln ln 2.5xe = Use the power property. 2 x ln ln 2.5e = Simplify using ln 1e = . 2 x ln 2.5=
Divide by 2. ln 2.5
x2
=
Use the calculator to finish. x 0.458≈ 8. Solve 1 30xe + = 9. Solve 34 1.2 14xe + =
Solve each equation. Round to the nearest ten-thousandth. Check your answer. Show all work for each. Circle your answer. 10. 18xe = 11. 2 12xe = 12. ln 3 6x =
13. ( )ln 4 1 36x − =
14. 5 4 7x
e + = 15. 22ln 2 x 1=
lne = 1
![Page 39: Practice 8-6 Natural Logarithmsstaff.bbhcsd.org/bradacm/files/2014/04/8.6-algebra-2.pdfApr 08, 2014 · Changing log e x y= to exponential form would give e xy =. Evaluating log e](https://reader035.fdocuments.in/reader035/viewer/2022070911/5fb18f72f20cc62f47153453/html5/thumbnails/39.jpg)
Name _____________________________ Class ___________________Date ____________
Practice 8-6 Natural Logarithms Remember that common logarithms are logarithms of base 10. 4 4
10log3 log 3x x+ +=
e is the base of the Natural Logarithms, often abbreviated as ln. ( ) ( )log ln xe x =
Often called Euler’s number, e is an irrational that has a value of 2.718281828459045…
Changing loge x y= to exponential form would give ye x= .
Evaluating loge x There are two keys on the TI-84 that are for evaluating exponential functions with base e. They are the second function of the / (e) and L (ex) keys. Evaluate each expression to the nearest thousandth.
1. 5e ________________ 2. 4e− ________________ 3. 1
3e ________________ The same properties (product, quotient and power) of exponents apply to natural logarithms. So 33ln ln lnx y x y+ = Write as a single natural logarithm.
4. 4 ln 2 ln f− = ________________ 5. 1
lnx 3ln2
y+ = ________________
Solving natural logarithmic equations.
Solve ( )2ln 3 5 4x + =
Write in exponential form. ( )24 3 5e x= +
Take the square root of both sides. 4 3 5e x± = +
Subtract 5 from both sides. 4 5 3e x± − =
Divide both sides by 3. 4 5
3
ex
± − =
Evaluate using the calculator. 7.39 and 4.130x x= − =
6. Solve ln 9 5x = 7. Solve 2
ln 123
x + =
![Page 40: Practice 8-6 Natural Logarithmsstaff.bbhcsd.org/bradacm/files/2014/04/8.6-algebra-2.pdfApr 08, 2014 · Changing log e x y= to exponential form would give e xy =. Evaluating log e](https://reader035.fdocuments.in/reader035/viewer/2022070911/5fb18f72f20cc62f47153453/html5/thumbnails/40.jpg)
Solving an exponential equation. Solve 27 2.5 20xe + = Subtract 2.5 from both sides to start 27 17.5xe = isolating the exponential term. Divide both sides by 7 to finish 2 2.5xe = isolating the exponential term. Take the natural log of both sides. 2ln ln 2.5xe = Use the power property. 2 x ln ln 2.5e = Simplify using ln 1e = . 2 x ln 2.5=
Divide by 2. ln 2.5
x2
=
Use the calculator to finish. x 0.458≈ 8. Solve 1 30xe + = 9. Solve 34 1.2 14xe + =
Solve each equation. Round to the nearest ten-thousandth. Check your answer. Show all work for each. Circle your answer. 10. 18xe = 11. 2 12xe = 12. ln 3 6x =
13. ( )ln 4 1 36x − =
14. 5 4 7x
e + = 15. 22ln 2 x 1=
lne = 1
![Page 41: Practice 8-6 Natural Logarithmsstaff.bbhcsd.org/bradacm/files/2014/04/8.6-algebra-2.pdfApr 08, 2014 · Changing log e x y= to exponential form would give e xy =. Evaluating log e](https://reader035.fdocuments.in/reader035/viewer/2022070911/5fb18f72f20cc62f47153453/html5/thumbnails/41.jpg)
Name _____________________________ Class ___________________Date ____________
Practice 8-6 Natural Logarithms Remember that common logarithms are logarithms of base 10. 4 4
10log3 log 3x x+ +=
e is the base of the Natural Logarithms, often abbreviated as ln. ( ) ( )log ln xe x =
Often called Euler’s number, e is an irrational that has a value of 2.718281828459045…
Changing loge x y= to exponential form would give ye x= .
Evaluating loge x There are two keys on the TI-84 that are for evaluating exponential functions with base e. They are the second function of the / (e) and L (ex) keys. Evaluate each expression to the nearest thousandth.
1. 5e ________________ 2. 4e− ________________ 3. 1
3e ________________ The same properties (product, quotient and power) of exponents apply to natural logarithms. So 33ln ln lnx y x y+ = Write as a single natural logarithm.
4. 4 ln 2 ln f− = ________________ 5. 1
lnx 3ln2
y+ = ________________
Solving natural logarithmic equations.
Solve ( )2ln 3 5 4x + =
Write in exponential form. ( )24 3 5e x= +
Take the square root of both sides. 4 3 5e x± = +
Subtract 5 from both sides. 4 5 3e x± − =
Divide both sides by 3. 4 5
3
ex
± − =
Evaluate using the calculator. 7.39 and 4.130x x= − =
6. Solve ln 9 5x = 7. Solve 2
ln 123
x + =
![Page 42: Practice 8-6 Natural Logarithmsstaff.bbhcsd.org/bradacm/files/2014/04/8.6-algebra-2.pdfApr 08, 2014 · Changing log e x y= to exponential form would give e xy =. Evaluating log e](https://reader035.fdocuments.in/reader035/viewer/2022070911/5fb18f72f20cc62f47153453/html5/thumbnails/42.jpg)
Solving an exponential equation. Solve 27 2.5 20xe + = Subtract 2.5 from both sides to start 27 17.5xe = isolating the exponential term. Divide both sides by 7 to finish 2 2.5xe = isolating the exponential term. Take the natural log of both sides. 2ln ln 2.5xe = Use the power property. 2 x ln ln 2.5e = Simplify using ln 1e = . 2 x ln 2.5=
Divide by 2. ln 2.5
x2
=
Use the calculator to finish. x 0.458≈ 8. Solve 1 30xe + = 9. Solve 34 1.2 14xe + =
Solve each equation. Round to the nearest ten-thousandth. Check your answer. Show all work for each. Circle your answer. 10. 18xe = 11. 2 12xe = 12. ln 3 6x =
13. ( )ln 4 1 36x − =
14. 5 4 7x
e + = 15. 22ln 2 x 1=
lne = 1
![Page 43: Practice 8-6 Natural Logarithmsstaff.bbhcsd.org/bradacm/files/2014/04/8.6-algebra-2.pdfApr 08, 2014 · Changing log e x y= to exponential form would give e xy =. Evaluating log e](https://reader035.fdocuments.in/reader035/viewer/2022070911/5fb18f72f20cc62f47153453/html5/thumbnails/43.jpg)
Name _____________________________ Class ___________________Date ____________
Practice 8-6 Natural Logarithms Remember that common logarithms are logarithms of base 10. 4 4
10log3 log 3x x+ +=
e is the base of the Natural Logarithms, often abbreviated as ln. ( ) ( )log ln xe x =
Often called Euler’s number, e is an irrational that has a value of 2.718281828459045…
Changing loge x y= to exponential form would give ye x= .
Evaluating loge x There are two keys on the TI-84 that are for evaluating exponential functions with base e. They are the second function of the / (e) and L (ex) keys. Evaluate each expression to the nearest thousandth.
1. 5e ________________ 2. 4e− ________________ 3. 1
3e ________________ The same properties (product, quotient and power) of exponents apply to natural logarithms. So 33ln ln lnx y x y+ = Write as a single natural logarithm.
4. 4 ln 2 ln f− = ________________ 5. 1
lnx 3ln2
y+ = ________________
Solving natural logarithmic equations.
Solve ( )2ln 3 5 4x + =
Write in exponential form. ( )24 3 5e x= +
Take the square root of both sides. 4 3 5e x± = +
Subtract 5 from both sides. 4 5 3e x± − =
Divide both sides by 3. 4 5
3
ex
± − =
Evaluate using the calculator. 7.39 and 4.130x x= − =
6. Solve ln 9 5x = 7. Solve 2
ln 123
x + =
![Page 44: Practice 8-6 Natural Logarithmsstaff.bbhcsd.org/bradacm/files/2014/04/8.6-algebra-2.pdfApr 08, 2014 · Changing log e x y= to exponential form would give e xy =. Evaluating log e](https://reader035.fdocuments.in/reader035/viewer/2022070911/5fb18f72f20cc62f47153453/html5/thumbnails/44.jpg)
Solving an exponential equation. Solve 27 2.5 20xe + = Subtract 2.5 from both sides to start 27 17.5xe = isolating the exponential term. Divide both sides by 7 to finish 2 2.5xe = isolating the exponential term. Take the natural log of both sides. 2ln ln 2.5xe = Use the power property. 2 x ln ln 2.5e = Simplify using ln 1e = . 2 x ln 2.5=
Divide by 2. ln 2.5
x2
=
Use the calculator to finish. x 0.458≈ 8. Solve 1 30xe + = 9. Solve 34 1.2 14xe + =
Solve each equation. Round to the nearest ten-thousandth. Check your answer. Show all work for each. Circle your answer. 10. 18xe = 11. 2 12xe = 12. ln 3 6x =
13. ( )ln 4 1 36x − =
14. 5 4 7x
e + = 15. 22ln 2 x 1=
lne = 1
![Page 45: Practice 8-6 Natural Logarithmsstaff.bbhcsd.org/bradacm/files/2014/04/8.6-algebra-2.pdfApr 08, 2014 · Changing log e x y= to exponential form would give e xy =. Evaluating log e](https://reader035.fdocuments.in/reader035/viewer/2022070911/5fb18f72f20cc62f47153453/html5/thumbnails/45.jpg)
Name _____________________________ Class ___________________Date ____________
Practice 8-6 Natural Logarithms Remember that common logarithms are logarithms of base 10. 4 4
10log3 log 3x x+ +=
e is the base of the Natural Logarithms, often abbreviated as ln. ( ) ( )log ln xe x =
Often called Euler’s number, e is an irrational that has a value of 2.718281828459045…
Changing loge x y= to exponential form would give ye x= .
Evaluating loge x There are two keys on the TI-84 that are for evaluating exponential functions with base e. They are the second function of the / (e) and L (ex) keys. Evaluate each expression to the nearest thousandth.
1. 5e ________________ 2. 4e− ________________ 3. 1
3e ________________ The same properties (product, quotient and power) of exponents apply to natural logarithms. So 33ln ln lnx y x y+ = Write as a single natural logarithm.
4. 4 ln 2 ln f− = ________________ 5. 1
lnx 3ln2
y+ = ________________
Solving natural logarithmic equations.
Solve ( )2ln 3 5 4x + =
Write in exponential form. ( )24 3 5e x= +
Take the square root of both sides. 4 3 5e x± = +
Subtract 5 from both sides. 4 5 3e x± − =
Divide both sides by 3. 4 5
3
ex
± − =
Evaluate using the calculator. 7.39 and 4.130x x= − =
6. Solve ln 9 5x = 7. Solve 2
ln 123
x + =
![Page 46: Practice 8-6 Natural Logarithmsstaff.bbhcsd.org/bradacm/files/2014/04/8.6-algebra-2.pdfApr 08, 2014 · Changing log e x y= to exponential form would give e xy =. Evaluating log e](https://reader035.fdocuments.in/reader035/viewer/2022070911/5fb18f72f20cc62f47153453/html5/thumbnails/46.jpg)
Solving an exponential equation. Solve 27 2.5 20xe + = Subtract 2.5 from both sides to start 27 17.5xe = isolating the exponential term. Divide both sides by 7 to finish 2 2.5xe = isolating the exponential term. Take the natural log of both sides. 2ln ln 2.5xe = Use the power property. 2 x ln ln 2.5e = Simplify using ln 1e = . 2 x ln 2.5=
Divide by 2. ln 2.5
x2
=
Use the calculator to finish. x 0.458≈ 8. Solve 1 30xe + = 9. Solve 34 1.2 14xe + =
Solve each equation. Round to the nearest ten-thousandth. Check your answer. Show all work for each. Circle your answer. 10. 18xe = 11. 2 12xe = 12. ln 3 6x =
13. ( )ln 4 1 36x − =
14. 5 4 7x
e + = 15. 22ln 2 x 1=
lne = 1
![Page 47: Practice 8-6 Natural Logarithmsstaff.bbhcsd.org/bradacm/files/2014/04/8.6-algebra-2.pdfApr 08, 2014 · Changing log e x y= to exponential form would give e xy =. Evaluating log e](https://reader035.fdocuments.in/reader035/viewer/2022070911/5fb18f72f20cc62f47153453/html5/thumbnails/47.jpg)
Name _____________________________ Class ___________________Date ____________
Practice 8-6 Natural Logarithms Remember that common logarithms are logarithms of base 10. 4 4
10log3 log 3x x+ +=
e is the base of the Natural Logarithms, often abbreviated as ln. ( ) ( )log ln xe x =
Often called Euler’s number, e is an irrational that has a value of 2.718281828459045…
Changing loge x y= to exponential form would give ye x= .
Evaluating loge x There are two keys on the TI-84 that are for evaluating exponential functions with base e. They are the second function of the / (e) and L (ex) keys. Evaluate each expression to the nearest thousandth.
1. 5e ________________ 2. 4e− ________________ 3. 1
3e ________________ The same properties (product, quotient and power) of exponents apply to natural logarithms. So 33ln ln lnx y x y+ = Write as a single natural logarithm.
4. 4 ln 2 ln f− = ________________ 5. 1
lnx 3ln2
y+ = ________________
Solving natural logarithmic equations.
Solve ( )2ln 3 5 4x + =
Write in exponential form. ( )24 3 5e x= +
Take the square root of both sides. 4 3 5e x± = +
Subtract 5 from both sides. 4 5 3e x± − =
Divide both sides by 3. 4 5
3
ex
± − =
Evaluate using the calculator. 7.39 and 4.130x x= − =
6. Solve ln 9 5x = 7. Solve 2
ln 123
x + =
![Page 48: Practice 8-6 Natural Logarithmsstaff.bbhcsd.org/bradacm/files/2014/04/8.6-algebra-2.pdfApr 08, 2014 · Changing log e x y= to exponential form would give e xy =. Evaluating log e](https://reader035.fdocuments.in/reader035/viewer/2022070911/5fb18f72f20cc62f47153453/html5/thumbnails/48.jpg)
Solving an exponential equation. Solve 27 2.5 20xe + = Subtract 2.5 from both sides to start 27 17.5xe = isolating the exponential term. Divide both sides by 7 to finish 2 2.5xe = isolating the exponential term. Take the natural log of both sides. 2ln ln 2.5xe = Use the power property. 2 x ln ln 2.5e = Simplify using ln 1e = . 2 x ln 2.5=
Divide by 2. ln 2.5
x2
=
Use the calculator to finish. x 0.458≈ 8. Solve 1 30xe + = 9. Solve 34 1.2 14xe + =
Solve each equation. Round to the nearest ten-thousandth. Check your answer. Show all work for each. Circle your answer. 10. 18xe = 11. 2 12xe = 12. ln 3 6x =
13. ( )ln 4 1 36x − =
14. 5 4 7x
e + = 15. 22ln 2 x 1=
lne = 1
![Page 49: Practice 8-6 Natural Logarithmsstaff.bbhcsd.org/bradacm/files/2014/04/8.6-algebra-2.pdfApr 08, 2014 · Changing log e x y= to exponential form would give e xy =. Evaluating log e](https://reader035.fdocuments.in/reader035/viewer/2022070911/5fb18f72f20cc62f47153453/html5/thumbnails/49.jpg)
Name _____________________________ Class ___________________Date ____________
Practice 8-6 Natural Logarithms Remember that common logarithms are logarithms of base 10. 4 4
10log3 log 3x x+ +=
e is the base of the Natural Logarithms, often abbreviated as ln. ( ) ( )log ln xe x =
Often called Euler’s number, e is an irrational that has a value of 2.718281828459045…
Changing loge x y= to exponential form would give ye x= .
Evaluating loge x There are two keys on the TI-84 that are for evaluating exponential functions with base e. They are the second function of the / (e) and L (ex) keys. Evaluate each expression to the nearest thousandth.
1. 5e ________________ 2. 4e− ________________ 3. 1
3e ________________ The same properties (product, quotient and power) of exponents apply to natural logarithms. So 33ln ln lnx y x y+ = Write as a single natural logarithm.
4. 4 ln 2 ln f− = ________________ 5. 1
lnx 3ln2
y+ = ________________
Solving natural logarithmic equations.
Solve ( )2ln 3 5 4x + =
Write in exponential form. ( )24 3 5e x= +
Take the square root of both sides. 4 3 5e x± = +
Subtract 5 from both sides. 4 5 3e x± − =
Divide both sides by 3. 4 5
3
ex
± − =
Evaluate using the calculator. 7.39 and 4.130x x= − =
6. Solve ln 9 5x = 7. Solve 2
ln 123
x + =
![Page 50: Practice 8-6 Natural Logarithmsstaff.bbhcsd.org/bradacm/files/2014/04/8.6-algebra-2.pdfApr 08, 2014 · Changing log e x y= to exponential form would give e xy =. Evaluating log e](https://reader035.fdocuments.in/reader035/viewer/2022070911/5fb18f72f20cc62f47153453/html5/thumbnails/50.jpg)
Solving an exponential equation. Solve 27 2.5 20xe + = Subtract 2.5 from both sides to start 27 17.5xe = isolating the exponential term. Divide both sides by 7 to finish 2 2.5xe = isolating the exponential term. Take the natural log of both sides. 2ln ln 2.5xe = Use the power property. 2 x ln ln 2.5e = Simplify using ln 1e = . 2 x ln 2.5=
Divide by 2. ln 2.5
x2
=
Use the calculator to finish. x 0.458≈ 8. Solve 1 30xe + = 9. Solve 34 1.2 14xe + =
Solve each equation. Round to the nearest ten-thousandth. Check your answer. Show all work for each. Circle your answer. 10. 18xe = 11. 2 12xe = 12. ln 3 6x =
13. ( )ln 4 1 36x − =
14. 5 4 7x
e + = 15. 22ln 2 x 1=
lne = 1
![Page 51: Practice 8-6 Natural Logarithmsstaff.bbhcsd.org/bradacm/files/2014/04/8.6-algebra-2.pdfApr 08, 2014 · Changing log e x y= to exponential form would give e xy =. Evaluating log e](https://reader035.fdocuments.in/reader035/viewer/2022070911/5fb18f72f20cc62f47153453/html5/thumbnails/51.jpg)
Name _____________________________ Class ___________________Date ____________
Practice 8-6 Natural Logarithms Remember that common logarithms are logarithms of base 10. 4 4
10log3 log 3x x+ +=
e is the base of the Natural Logarithms, often abbreviated as ln. ( ) ( )log ln xe x =
Often called Euler’s number, e is an irrational that has a value of 2.718281828459045…
Changing loge x y= to exponential form would give ye x= .
Evaluating loge x There are two keys on the TI-84 that are for evaluating exponential functions with base e. They are the second function of the / (e) and L (ex) keys. Evaluate each expression to the nearest thousandth.
1. 5e ________________ 2. 4e− ________________ 3. 1
3e ________________ The same properties (product, quotient and power) of exponents apply to natural logarithms. So 33ln ln lnx y x y+ = Write as a single natural logarithm.
4. 4 ln 2 ln f− = ________________ 5. 1
lnx 3ln2
y+ = ________________
Solving natural logarithmic equations.
Solve ( )2ln 3 5 4x + =
Write in exponential form. ( )24 3 5e x= +
Take the square root of both sides. 4 3 5e x± = +
Subtract 5 from both sides. 4 5 3e x± − =
Divide both sides by 3. 4 5
3
ex
± − =
Evaluate using the calculator. 7.39 and 4.130x x= − =
6. Solve ln 9 5x = 7. Solve 2
ln 123
x + =
![Page 52: Practice 8-6 Natural Logarithmsstaff.bbhcsd.org/bradacm/files/2014/04/8.6-algebra-2.pdfApr 08, 2014 · Changing log e x y= to exponential form would give e xy =. Evaluating log e](https://reader035.fdocuments.in/reader035/viewer/2022070911/5fb18f72f20cc62f47153453/html5/thumbnails/52.jpg)
Solving an exponential equation. Solve 27 2.5 20xe + = Subtract 2.5 from both sides to start 27 17.5xe = isolating the exponential term. Divide both sides by 7 to finish 2 2.5xe = isolating the exponential term. Take the natural log of both sides. 2ln ln 2.5xe = Use the power property. 2 x ln ln 2.5e = Simplify using ln 1e = . 2 x ln 2.5=
Divide by 2. ln 2.5
x2
=
Use the calculator to finish. x 0.458≈ 8. Solve 1 30xe + = 9. Solve 34 1.2 14xe + =
Solve each equation. Round to the nearest ten-thousandth. Check your answer. Show all work for each. Circle your answer. 10. 18xe = 11. 2 12xe = 12. ln 3 6x =
13. ( )ln 4 1 36x − =
14. 5 4 7x
e + = 15. 22ln 2 x 1=
lne = 1
![Page 53: Practice 8-6 Natural Logarithmsstaff.bbhcsd.org/bradacm/files/2014/04/8.6-algebra-2.pdfApr 08, 2014 · Changing log e x y= to exponential form would give e xy =. Evaluating log e](https://reader035.fdocuments.in/reader035/viewer/2022070911/5fb18f72f20cc62f47153453/html5/thumbnails/53.jpg)
Name _____________________________ Class ___________________Date ____________
Practice 8-6 Natural Logarithms Remember that common logarithms are logarithms of base 10. 4 4
10log3 log 3x x+ +=
e is the base of the Natural Logarithms, often abbreviated as ln. ( ) ( )log ln xe x =
Often called Euler’s number, e is an irrational that has a value of 2.718281828459045…
Changing loge x y= to exponential form would give ye x= .
Evaluating loge x There are two keys on the TI-84 that are for evaluating exponential functions with base e. They are the second function of the / (e) and L (ex) keys. Evaluate each expression to the nearest thousandth.
1. 5e ________________ 2. 4e− ________________ 3. 1
3e ________________ The same properties (product, quotient and power) of exponents apply to natural logarithms. So 33ln ln lnx y x y+ = Write as a single natural logarithm.
4. 4 ln 2 ln f− = ________________ 5. 1
lnx 3ln2
y+ = ________________
Solving natural logarithmic equations.
Solve ( )2ln 3 5 4x + =
Write in exponential form. ( )24 3 5e x= +
Take the square root of both sides. 4 3 5e x± = +
Subtract 5 from both sides. 4 5 3e x± − =
Divide both sides by 3. 4 5
3
ex
± − =
Evaluate using the calculator. 7.39 and 4.130x x= − =
6. Solve ln 9 5x = 7. Solve 2
ln 123
x + =
![Page 54: Practice 8-6 Natural Logarithmsstaff.bbhcsd.org/bradacm/files/2014/04/8.6-algebra-2.pdfApr 08, 2014 · Changing log e x y= to exponential form would give e xy =. Evaluating log e](https://reader035.fdocuments.in/reader035/viewer/2022070911/5fb18f72f20cc62f47153453/html5/thumbnails/54.jpg)
Solving an exponential equation. Solve 27 2.5 20xe + = Subtract 2.5 from both sides to start 27 17.5xe = isolating the exponential term. Divide both sides by 7 to finish 2 2.5xe = isolating the exponential term. Take the natural log of both sides. 2ln ln 2.5xe = Use the power property. 2 x ln ln 2.5e = Simplify using ln 1e = . 2 x ln 2.5=
Divide by 2. ln 2.5
x2
=
Use the calculator to finish. x 0.458≈ 8. Solve 1 30xe + = 9. Solve 34 1.2 14xe + =
Solve each equation. Round to the nearest ten-thousandth. Check your answer. Show all work for each. Circle your answer. 10. 18xe = 11. 2 12xe = 12. ln 3 6x =
13. ( )ln 4 1 36x − =
14. 5 4 7x
e + = 15. 22ln 2 x 1=
lne = 1
![Page 55: Practice 8-6 Natural Logarithmsstaff.bbhcsd.org/bradacm/files/2014/04/8.6-algebra-2.pdfApr 08, 2014 · Changing log e x y= to exponential form would give e xy =. Evaluating log e](https://reader035.fdocuments.in/reader035/viewer/2022070911/5fb18f72f20cc62f47153453/html5/thumbnails/55.jpg)
Name _____________________________ Class ___________________Date ____________
Practice 8-6 Natural Logarithms Remember that common logarithms are logarithms of base 10. 4 4
10log3 log 3x x+ +=
e is the base of the Natural Logarithms, often abbreviated as ln. ( ) ( )log ln xe x =
Often called Euler’s number, e is an irrational that has a value of 2.718281828459045…
Changing loge x y= to exponential form would give ye x= .
Evaluating loge x There are two keys on the TI-84 that are for evaluating exponential functions with base e. They are the second function of the / (e) and L (ex) keys. Evaluate each expression to the nearest thousandth.
1. 5e ________________ 2. 4e− ________________ 3. 1
3e ________________ The same properties (product, quotient and power) of exponents apply to natural logarithms. So 33ln ln lnx y x y+ = Write as a single natural logarithm.
4. 4 ln 2 ln f− = ________________ 5. 1
lnx 3ln2
y+ = ________________
Solving natural logarithmic equations.
Solve ( )2ln 3 5 4x + =
Write in exponential form. ( )24 3 5e x= +
Take the square root of both sides. 4 3 5e x± = +
Subtract 5 from both sides. 4 5 3e x± − =
Divide both sides by 3. 4 5
3
ex
± − =
Evaluate using the calculator. 7.39 and 4.130x x= − =
6. Solve ln 9 5x = 7. Solve 2
ln 123
x + =
![Page 56: Practice 8-6 Natural Logarithmsstaff.bbhcsd.org/bradacm/files/2014/04/8.6-algebra-2.pdfApr 08, 2014 · Changing log e x y= to exponential form would give e xy =. Evaluating log e](https://reader035.fdocuments.in/reader035/viewer/2022070911/5fb18f72f20cc62f47153453/html5/thumbnails/56.jpg)
Solving an exponential equation. Solve 27 2.5 20xe + = Subtract 2.5 from both sides to start 27 17.5xe = isolating the exponential term. Divide both sides by 7 to finish 2 2.5xe = isolating the exponential term. Take the natural log of both sides. 2ln ln 2.5xe = Use the power property. 2 x ln ln 2.5e = Simplify using ln 1e = . 2 x ln 2.5=
Divide by 2. ln 2.5
x2
=
Use the calculator to finish. x 0.458≈ 8. Solve 1 30xe + = 9. Solve 34 1.2 14xe + =
Solve each equation. Round to the nearest ten-thousandth. Check your answer. Show all work for each. Circle your answer. 10. 18xe = 11. 2 12xe = 12. ln 3 6x =
13. ( )ln 4 1 36x − =
14. 5 4 7x
e + = 15. 22ln 2 x 1=
lne = 1
![Page 57: Practice 8-6 Natural Logarithmsstaff.bbhcsd.org/bradacm/files/2014/04/8.6-algebra-2.pdfApr 08, 2014 · Changing log e x y= to exponential form would give e xy =. Evaluating log e](https://reader035.fdocuments.in/reader035/viewer/2022070911/5fb18f72f20cc62f47153453/html5/thumbnails/57.jpg)
Name _____________________________ Class ___________________Date ____________
Practice 8-6 Natural Logarithms Remember that common logarithms are logarithms of base 10. 4 4
10log3 log 3x x+ +=
e is the base of the Natural Logarithms, often abbreviated as ln. ( ) ( )log ln xe x =
Often called Euler’s number, e is an irrational that has a value of 2.718281828459045…
Changing loge x y= to exponential form would give ye x= .
Evaluating loge x There are two keys on the TI-84 that are for evaluating exponential functions with base e. They are the second function of the / (e) and L (ex) keys. Evaluate each expression to the nearest thousandth.
1. 5e ________________ 2. 4e− ________________ 3. 1
3e ________________ The same properties (product, quotient and power) of exponents apply to natural logarithms. So 33ln ln lnx y x y+ = Write as a single natural logarithm.
4. 4 ln 2 ln f− = ________________ 5. 1
lnx 3ln2
y+ = ________________
Solving natural logarithmic equations.
Solve ( )2ln 3 5 4x + =
Write in exponential form. ( )24 3 5e x= +
Take the square root of both sides. 4 3 5e x± = +
Subtract 5 from both sides. 4 5 3e x± − =
Divide both sides by 3. 4 5
3
ex
± − =
Evaluate using the calculator. 7.39 and 4.130x x= − =
6. Solve ln 9 5x = 7. Solve 2
ln 123
x + =
![Page 58: Practice 8-6 Natural Logarithmsstaff.bbhcsd.org/bradacm/files/2014/04/8.6-algebra-2.pdfApr 08, 2014 · Changing log e x y= to exponential form would give e xy =. Evaluating log e](https://reader035.fdocuments.in/reader035/viewer/2022070911/5fb18f72f20cc62f47153453/html5/thumbnails/58.jpg)
Solving an exponential equation. Solve 27 2.5 20xe + = Subtract 2.5 from both sides to start 27 17.5xe = isolating the exponential term. Divide both sides by 7 to finish 2 2.5xe = isolating the exponential term. Take the natural log of both sides. 2ln ln 2.5xe = Use the power property. 2 x ln ln 2.5e = Simplify using ln 1e = . 2 x ln 2.5=
Divide by 2. ln 2.5
x2
=
Use the calculator to finish. x 0.458≈ 8. Solve 1 30xe + = 9. Solve 34 1.2 14xe + =
Solve each equation. Round to the nearest ten-thousandth. Check your answer. Show all work for each. Circle your answer. 10. 18xe = 11. 2 12xe = 12. ln 3 6x =
13. ( )ln 4 1 36x − =
14. 5 4 7x
e + = 15. 22ln 2 x 1=
lne = 1
![Page 59: Practice 8-6 Natural Logarithmsstaff.bbhcsd.org/bradacm/files/2014/04/8.6-algebra-2.pdfApr 08, 2014 · Changing log e x y= to exponential form would give e xy =. Evaluating log e](https://reader035.fdocuments.in/reader035/viewer/2022070911/5fb18f72f20cc62f47153453/html5/thumbnails/59.jpg)
Name _____________________________ Class ___________________Date ____________
Practice 8-6 Natural Logarithms Remember that common logarithms are logarithms of base 10. 4 4
10log3 log 3x x+ +=
e is the base of the Natural Logarithms, often abbreviated as ln. ( ) ( )log ln xe x =
Often called Euler’s number, e is an irrational that has a value of 2.718281828459045…
Changing loge x y= to exponential form would give ye x= .
Evaluating loge x There are two keys on the TI-84 that are for evaluating exponential functions with base e. They are the second function of the / (e) and L (ex) keys. Evaluate each expression to the nearest thousandth.
1. 5e ________________ 2. 4e− ________________ 3. 1
3e ________________ The same properties (product, quotient and power) of exponents apply to natural logarithms. So 33ln ln lnx y x y+ = Write as a single natural logarithm.
4. 4 ln 2 ln f− = ________________ 5. 1
lnx 3ln2
y+ = ________________
Solving natural logarithmic equations.
Solve ( )2ln 3 5 4x + =
Write in exponential form. ( )24 3 5e x= +
Take the square root of both sides. 4 3 5e x± = +
Subtract 5 from both sides. 4 5 3e x± − =
Divide both sides by 3. 4 5
3
ex
± − =
Evaluate using the calculator. 7.39 and 4.130x x= − =
6. Solve ln 9 5x = 7. Solve 2
ln 123
x + =
![Page 60: Practice 8-6 Natural Logarithmsstaff.bbhcsd.org/bradacm/files/2014/04/8.6-algebra-2.pdfApr 08, 2014 · Changing log e x y= to exponential form would give e xy =. Evaluating log e](https://reader035.fdocuments.in/reader035/viewer/2022070911/5fb18f72f20cc62f47153453/html5/thumbnails/60.jpg)
Solving an exponential equation. Solve 27 2.5 20xe + = Subtract 2.5 from both sides to start 27 17.5xe = isolating the exponential term. Divide both sides by 7 to finish 2 2.5xe = isolating the exponential term. Take the natural log of both sides. 2ln ln 2.5xe = Use the power property. 2 x ln ln 2.5e = Simplify using ln 1e = . 2 x ln 2.5=
Divide by 2. ln 2.5
x2
=
Use the calculator to finish. x 0.458≈ 8. Solve 1 30xe + = 9. Solve 34 1.2 14xe + =
Solve each equation. Round to the nearest ten-thousandth. Check your answer. Show all work for each. Circle your answer. 10. 18xe = 11. 2 12xe = 12. ln 3 6x =
13. ( )ln 4 1 36x − =
14. 5 4 7x
e + = 15. 22ln 2 x 1=
lne = 1
![Page 61: Practice 8-6 Natural Logarithmsstaff.bbhcsd.org/bradacm/files/2014/04/8.6-algebra-2.pdfApr 08, 2014 · Changing log e x y= to exponential form would give e xy =. Evaluating log e](https://reader035.fdocuments.in/reader035/viewer/2022070911/5fb18f72f20cc62f47153453/html5/thumbnails/61.jpg)
Name _____________________________ Class ___________________Date ____________
Practice 8-6 Natural Logarithms Remember that common logarithms are logarithms of base 10. 4 4
10log3 log 3x x+ +=
e is the base of the Natural Logarithms, often abbreviated as ln. ( ) ( )log ln xe x =
Often called Euler’s number, e is an irrational that has a value of 2.718281828459045…
Changing loge x y= to exponential form would give ye x= .
Evaluating loge x There are two keys on the TI-84 that are for evaluating exponential functions with base e. They are the second function of the / (e) and L (ex) keys. Evaluate each expression to the nearest thousandth.
1. 5e ________________ 2. 4e− ________________ 3. 1
3e ________________ The same properties (product, quotient and power) of exponents apply to natural logarithms. So 33ln ln lnx y x y+ = Write as a single natural logarithm.
4. 4 ln 2 ln f− = ________________ 5. 1
lnx 3ln2
y+ = ________________
Solving natural logarithmic equations.
Solve ( )2ln 3 5 4x + =
Write in exponential form. ( )24 3 5e x= +
Take the square root of both sides. 4 3 5e x± = +
Subtract 5 from both sides. 4 5 3e x± − =
Divide both sides by 3. 4 5
3
ex
± − =
Evaluate using the calculator. 7.39 and 4.130x x= − =
6. Solve ln 9 5x = 7. Solve 2
ln 123
x + =
![Page 62: Practice 8-6 Natural Logarithmsstaff.bbhcsd.org/bradacm/files/2014/04/8.6-algebra-2.pdfApr 08, 2014 · Changing log e x y= to exponential form would give e xy =. Evaluating log e](https://reader035.fdocuments.in/reader035/viewer/2022070911/5fb18f72f20cc62f47153453/html5/thumbnails/62.jpg)
Solving an exponential equation. Solve 27 2.5 20xe + = Subtract 2.5 from both sides to start 27 17.5xe = isolating the exponential term. Divide both sides by 7 to finish 2 2.5xe = isolating the exponential term. Take the natural log of both sides. 2ln ln 2.5xe = Use the power property. 2 x ln ln 2.5e = Simplify using ln 1e = . 2 x ln 2.5=
Divide by 2. ln 2.5
x2
=
Use the calculator to finish. x 0.458≈ 8. Solve 1 30xe + = 9. Solve 34 1.2 14xe + =
Solve each equation. Round to the nearest ten-thousandth. Check your answer. Show all work for each. Circle your answer. 10. 18xe = 11. 2 12xe = 12. ln 3 6x =
13. ( )ln 4 1 36x − =
14. 5 4 7x
e + = 15. 22ln 2 x 1=
lne = 1
![Page 63: Practice 8-6 Natural Logarithmsstaff.bbhcsd.org/bradacm/files/2014/04/8.6-algebra-2.pdfApr 08, 2014 · Changing log e x y= to exponential form would give e xy =. Evaluating log e](https://reader035.fdocuments.in/reader035/viewer/2022070911/5fb18f72f20cc62f47153453/html5/thumbnails/63.jpg)
Name _____________________________ Class ___________________Date ____________
Practice 8-6 Natural Logarithms Remember that common logarithms are logarithms of base 10. 4 4
10log3 log 3x x+ +=
e is the base of the Natural Logarithms, often abbreviated as ln. ( ) ( )log ln xe x =
Often called Euler’s number, e is an irrational that has a value of 2.718281828459045…
Changing loge x y= to exponential form would give ye x= .
Evaluating loge x There are two keys on the TI-84 that are for evaluating exponential functions with base e. They are the second function of the / (e) and L (ex) keys. Evaluate each expression to the nearest thousandth.
1. 5e ________________ 2. 4e− ________________ 3. 1
3e ________________ The same properties (product, quotient and power) of exponents apply to natural logarithms. So 33ln ln lnx y x y+ = Write as a single natural logarithm.
4. 4 ln 2 ln f− = ________________ 5. 1
lnx 3ln2
y+ = ________________
Solving natural logarithmic equations.
Solve ( )2ln 3 5 4x + =
Write in exponential form. ( )24 3 5e x= +
Take the square root of both sides. 4 3 5e x± = +
Subtract 5 from both sides. 4 5 3e x± − =
Divide both sides by 3. 4 5
3
ex
± − =
Evaluate using the calculator. 7.39 and 4.130x x= − =
6. Solve ln 9 5x = 7. Solve 2
ln 123
x + =
![Page 64: Practice 8-6 Natural Logarithmsstaff.bbhcsd.org/bradacm/files/2014/04/8.6-algebra-2.pdfApr 08, 2014 · Changing log e x y= to exponential form would give e xy =. Evaluating log e](https://reader035.fdocuments.in/reader035/viewer/2022070911/5fb18f72f20cc62f47153453/html5/thumbnails/64.jpg)
Solving an exponential equation. Solve 27 2.5 20xe + = Subtract 2.5 from both sides to start 27 17.5xe = isolating the exponential term. Divide both sides by 7 to finish 2 2.5xe = isolating the exponential term. Take the natural log of both sides. 2ln ln 2.5xe = Use the power property. 2 x ln ln 2.5e = Simplify using ln 1e = . 2 x ln 2.5=
Divide by 2. ln 2.5
x2
=
Use the calculator to finish. x 0.458≈ 8. Solve 1 30xe + = 9. Solve 34 1.2 14xe + =
Solve each equation. Round to the nearest ten-thousandth. Check your answer. Show all work for each. Circle your answer. 10. 18xe = 11. 2 12xe = 12. ln 3 6x =
13. ( )ln 4 1 36x − =
14. 5 4 7x
e + = 15. 22ln 2 x 1=
lne = 1
![Page 65: Practice 8-6 Natural Logarithmsstaff.bbhcsd.org/bradacm/files/2014/04/8.6-algebra-2.pdfApr 08, 2014 · Changing log e x y= to exponential form would give e xy =. Evaluating log e](https://reader035.fdocuments.in/reader035/viewer/2022070911/5fb18f72f20cc62f47153453/html5/thumbnails/65.jpg)
Name _____________________________ Class ___________________Date ____________
Practice 8-6 Natural Logarithms Remember that common logarithms are logarithms of base 10. 4 4
10log3 log 3x x+ +=
e is the base of the Natural Logarithms, often abbreviated as ln. ( ) ( )log ln xe x =
Often called Euler’s number, e is an irrational that has a value of 2.718281828459045…
Changing loge x y= to exponential form would give ye x= .
Evaluating loge x There are two keys on the TI-84 that are for evaluating exponential functions with base e. They are the second function of the / (e) and L (ex) keys. Evaluate each expression to the nearest thousandth.
1. 5e ________________ 2. 4e− ________________ 3. 1
3e ________________ The same properties (product, quotient and power) of exponents apply to natural logarithms. So 33ln ln lnx y x y+ = Write as a single natural logarithm.
4. 4 ln 2 ln f− = ________________ 5. 1
lnx 3ln2
y+ = ________________
Solving natural logarithmic equations.
Solve ( )2ln 3 5 4x + =
Write in exponential form. ( )24 3 5e x= +
Take the square root of both sides. 4 3 5e x± = +
Subtract 5 from both sides. 4 5 3e x± − =
Divide both sides by 3. 4 5
3
ex
± − =
Evaluate using the calculator. 7.39 and 4.130x x= − =
6. Solve ln 9 5x = 7. Solve 2
ln 123
x + =
![Page 66: Practice 8-6 Natural Logarithmsstaff.bbhcsd.org/bradacm/files/2014/04/8.6-algebra-2.pdfApr 08, 2014 · Changing log e x y= to exponential form would give e xy =. Evaluating log e](https://reader035.fdocuments.in/reader035/viewer/2022070911/5fb18f72f20cc62f47153453/html5/thumbnails/66.jpg)
Solving an exponential equation. Solve 27 2.5 20xe + = Subtract 2.5 from both sides to start 27 17.5xe = isolating the exponential term. Divide both sides by 7 to finish 2 2.5xe = isolating the exponential term. Take the natural log of both sides. 2ln ln 2.5xe = Use the power property. 2 x ln ln 2.5e = Simplify using ln 1e = . 2 x ln 2.5=
Divide by 2. ln 2.5
x2
=
Use the calculator to finish. x 0.458≈ 8. Solve 1 30xe + = 9. Solve 34 1.2 14xe + =
Solve each equation. Round to the nearest ten-thousandth. Check your answer. Show all work for each. Circle your answer. 10. 18xe = 11. 2 12xe = 12. ln 3 6x =
13. ( )ln 4 1 36x − =
14. 5 4 7x
e + = 15. 22ln 2 x 1=
lne = 1
![Page 67: Practice 8-6 Natural Logarithmsstaff.bbhcsd.org/bradacm/files/2014/04/8.6-algebra-2.pdfApr 08, 2014 · Changing log e x y= to exponential form would give e xy =. Evaluating log e](https://reader035.fdocuments.in/reader035/viewer/2022070911/5fb18f72f20cc62f47153453/html5/thumbnails/67.jpg)
Name _____________________________ Class ___________________Date ____________
Practice 8-6 Natural Logarithms Remember that common logarithms are logarithms of base 10. 4 4
10log3 log 3x x+ +=
e is the base of the Natural Logarithms, often abbreviated as ln. ( ) ( )log ln xe x =
Often called Euler’s number, e is an irrational that has a value of 2.718281828459045…
Changing loge x y= to exponential form would give ye x= .
Evaluating loge x There are two keys on the TI-84 that are for evaluating exponential functions with base e. They are the second function of the / (e) and L (ex) keys. Evaluate each expression to the nearest thousandth.
1. 5e ________________ 2. 4e− ________________ 3. 1
3e ________________ The same properties (product, quotient and power) of exponents apply to natural logarithms. So 33ln ln lnx y x y+ = Write as a single natural logarithm.
4. 4 ln 2 ln f− = ________________ 5. 1
lnx 3ln2
y+ = ________________
Solving natural logarithmic equations.
Solve ( )2ln 3 5 4x + =
Write in exponential form. ( )24 3 5e x= +
Take the square root of both sides. 4 3 5e x± = +
Subtract 5 from both sides. 4 5 3e x± − =
Divide both sides by 3. 4 5
3
ex
± − =
Evaluate using the calculator. 7.39 and 4.130x x= − =
6. Solve ln 9 5x = 7. Solve 2
ln 123
x + =
![Page 68: Practice 8-6 Natural Logarithmsstaff.bbhcsd.org/bradacm/files/2014/04/8.6-algebra-2.pdfApr 08, 2014 · Changing log e x y= to exponential form would give e xy =. Evaluating log e](https://reader035.fdocuments.in/reader035/viewer/2022070911/5fb18f72f20cc62f47153453/html5/thumbnails/68.jpg)
Solving an exponential equation. Solve 27 2.5 20xe + = Subtract 2.5 from both sides to start 27 17.5xe = isolating the exponential term. Divide both sides by 7 to finish 2 2.5xe = isolating the exponential term. Take the natural log of both sides. 2ln ln 2.5xe = Use the power property. 2 x ln ln 2.5e = Simplify using ln 1e = . 2 x ln 2.5=
Divide by 2. ln 2.5
x2
=
Use the calculator to finish. x 0.458≈ 8. Solve 1 30xe + = 9. Solve 34 1.2 14xe + =
Solve each equation. Round to the nearest ten-thousandth. Check your answer. Show all work for each. Circle your answer. 10. 18xe = 11. 2 12xe = 12. ln 3 6x =
13. ( )ln 4 1 36x − =
14. 5 4 7x
e + = 15. 22ln 2 x 1=
lne = 1
![Page 69: Practice 8-6 Natural Logarithmsstaff.bbhcsd.org/bradacm/files/2014/04/8.6-algebra-2.pdfApr 08, 2014 · Changing log e x y= to exponential form would give e xy =. Evaluating log e](https://reader035.fdocuments.in/reader035/viewer/2022070911/5fb18f72f20cc62f47153453/html5/thumbnails/69.jpg)
Name _____________________________ Class ___________________Date ____________
Practice 8-6 Natural Logarithms Remember that common logarithms are logarithms of base 10. 4 4
10log3 log 3x x+ +=
e is the base of the Natural Logarithms, often abbreviated as ln. ( ) ( )log ln xe x =
Often called Euler’s number, e is an irrational that has a value of 2.718281828459045…
Changing loge x y= to exponential form would give ye x= .
Evaluating loge x There are two keys on the TI-84 that are for evaluating exponential functions with base e. They are the second function of the / (e) and L (ex) keys. Evaluate each expression to the nearest thousandth.
1. 5e ________________ 2. 4e− ________________ 3. 1
3e ________________ The same properties (product, quotient and power) of exponents apply to natural logarithms. So 33ln ln lnx y x y+ = Write as a single natural logarithm.
4. 4 ln 2 ln f− = ________________ 5. 1
lnx 3ln2
y+ = ________________
Solving natural logarithmic equations.
Solve ( )2ln 3 5 4x + =
Write in exponential form. ( )24 3 5e x= +
Take the square root of both sides. 4 3 5e x± = +
Subtract 5 from both sides. 4 5 3e x± − =
Divide both sides by 3. 4 5
3
ex
± − =
Evaluate using the calculator. 7.39 and 4.130x x= − =
6. Solve ln 9 5x = 7. Solve 2
ln 123
x + =
![Page 70: Practice 8-6 Natural Logarithmsstaff.bbhcsd.org/bradacm/files/2014/04/8.6-algebra-2.pdfApr 08, 2014 · Changing log e x y= to exponential form would give e xy =. Evaluating log e](https://reader035.fdocuments.in/reader035/viewer/2022070911/5fb18f72f20cc62f47153453/html5/thumbnails/70.jpg)
Solving an exponential equation. Solve 27 2.5 20xe + = Subtract 2.5 from both sides to start 27 17.5xe = isolating the exponential term. Divide both sides by 7 to finish 2 2.5xe = isolating the exponential term. Take the natural log of both sides. 2ln ln 2.5xe = Use the power property. 2 x ln ln 2.5e = Simplify using ln 1e = . 2 x ln 2.5=
Divide by 2. ln 2.5
x2
=
Use the calculator to finish. x 0.458≈ 8. Solve 1 30xe + = 9. Solve 34 1.2 14xe + =
Solve each equation. Round to the nearest ten-thousandth. Check your answer. Show all work for each. Circle your answer. 10. 18xe = 11. 2 12xe = 12. ln 3 6x =
13. ( )ln 4 1 36x − =
14. 5 4 7x
e + = 15. 22ln 2 x 1=
lne = 1
![Page 71: Practice 8-6 Natural Logarithmsstaff.bbhcsd.org/bradacm/files/2014/04/8.6-algebra-2.pdfApr 08, 2014 · Changing log e x y= to exponential form would give e xy =. Evaluating log e](https://reader035.fdocuments.in/reader035/viewer/2022070911/5fb18f72f20cc62f47153453/html5/thumbnails/71.jpg)
Name _____________________________ Class ___________________Date ____________
Practice 8-6 Natural Logarithms Remember that common logarithms are logarithms of base 10. 4 4
10log3 log 3x x+ +=
e is the base of the Natural Logarithms, often abbreviated as ln. ( ) ( )log ln xe x =
Often called Euler’s number, e is an irrational that has a value of 2.718281828459045…
Changing loge x y= to exponential form would give ye x= .
Evaluating loge x There are two keys on the TI-84 that are for evaluating exponential functions with base e. They are the second function of the / (e) and L (ex) keys. Evaluate each expression to the nearest thousandth.
1. 5e ________________ 2. 4e− ________________ 3. 1
3e ________________ The same properties (product, quotient and power) of exponents apply to natural logarithms. So 33ln ln lnx y x y+ = Write as a single natural logarithm.
4. 4 ln 2 ln f− = ________________ 5. 1
lnx 3ln2
y+ = ________________
Solving natural logarithmic equations.
Solve ( )2ln 3 5 4x + =
Write in exponential form. ( )24 3 5e x= +
Take the square root of both sides. 4 3 5e x± = +
Subtract 5 from both sides. 4 5 3e x± − =
Divide both sides by 3. 4 5
3
ex
± − =
Evaluate using the calculator. 7.39 and 4.130x x= − =
6. Solve ln 9 5x = 7. Solve 2
ln 123
x + =
![Page 72: Practice 8-6 Natural Logarithmsstaff.bbhcsd.org/bradacm/files/2014/04/8.6-algebra-2.pdfApr 08, 2014 · Changing log e x y= to exponential form would give e xy =. Evaluating log e](https://reader035.fdocuments.in/reader035/viewer/2022070911/5fb18f72f20cc62f47153453/html5/thumbnails/72.jpg)
Solving an exponential equation. Solve 27 2.5 20xe + = Subtract 2.5 from both sides to start 27 17.5xe = isolating the exponential term. Divide both sides by 7 to finish 2 2.5xe = isolating the exponential term. Take the natural log of both sides. 2ln ln 2.5xe = Use the power property. 2 x ln ln 2.5e = Simplify using ln 1e = . 2 x ln 2.5=
Divide by 2. ln 2.5
x2
=
Use the calculator to finish. x 0.458≈ 8. Solve 1 30xe + = 9. Solve 34 1.2 14xe + =
Solve each equation. Round to the nearest ten-thousandth. Check your answer. Show all work for each. Circle your answer. 10. 18xe = 11. 2 12xe = 12. ln 3 6x =
13. ( )ln 4 1 36x − =
14. 5 4 7x
e + = 15. 22ln 2 x 1=
lne = 1
![Page 73: Practice 8-6 Natural Logarithmsstaff.bbhcsd.org/bradacm/files/2014/04/8.6-algebra-2.pdfApr 08, 2014 · Changing log e x y= to exponential form would give e xy =. Evaluating log e](https://reader035.fdocuments.in/reader035/viewer/2022070911/5fb18f72f20cc62f47153453/html5/thumbnails/73.jpg)
Name _____________________________ Class ___________________Date ____________
Practice 8-6 Natural Logarithms Remember that common logarithms are logarithms of base 10. 4 4
10log3 log 3x x+ +=
e is the base of the Natural Logarithms, often abbreviated as ln. ( ) ( )log ln xe x =
Often called Euler’s number, e is an irrational that has a value of 2.718281828459045…
Changing loge x y= to exponential form would give ye x= .
Evaluating loge x There are two keys on the TI-84 that are for evaluating exponential functions with base e. They are the second function of the / (e) and L (ex) keys. Evaluate each expression to the nearest thousandth.
1. 5e ________________ 2. 4e− ________________ 3. 1
3e ________________ The same properties (product, quotient and power) of exponents apply to natural logarithms. So 33ln ln lnx y x y+ = Write as a single natural logarithm.
4. 4 ln 2 ln f− = ________________ 5. 1
lnx 3ln2
y+ = ________________
Solving natural logarithmic equations.
Solve ( )2ln 3 5 4x + =
Write in exponential form. ( )24 3 5e x= +
Take the square root of both sides. 4 3 5e x± = +
Subtract 5 from both sides. 4 5 3e x± − =
Divide both sides by 3. 4 5
3
ex
± − =
Evaluate using the calculator. 7.39 and 4.130x x= − =
6. Solve ln 9 5x = 7. Solve 2
ln 123
x + =
![Page 74: Practice 8-6 Natural Logarithmsstaff.bbhcsd.org/bradacm/files/2014/04/8.6-algebra-2.pdfApr 08, 2014 · Changing log e x y= to exponential form would give e xy =. Evaluating log e](https://reader035.fdocuments.in/reader035/viewer/2022070911/5fb18f72f20cc62f47153453/html5/thumbnails/74.jpg)
Solving an exponential equation. Solve 27 2.5 20xe + = Subtract 2.5 from both sides to start 27 17.5xe = isolating the exponential term. Divide both sides by 7 to finish 2 2.5xe = isolating the exponential term. Take the natural log of both sides. 2ln ln 2.5xe = Use the power property. 2 x ln ln 2.5e = Simplify using ln 1e = . 2 x ln 2.5=
Divide by 2. ln 2.5
x2
=
Use the calculator to finish. x 0.458≈ 8. Solve 1 30xe + = 9. Solve 34 1.2 14xe + =
Solve each equation. Round to the nearest ten-thousandth. Check your answer. Show all work for each. Circle your answer. 10. 18xe = 11. 2 12xe = 12. ln 3 6x =
13. ( )ln 4 1 36x − =
14. 5 4 7x
e + = 15. 22ln 2 x 1=
lne = 1
![Page 75: Practice 8-6 Natural Logarithmsstaff.bbhcsd.org/bradacm/files/2014/04/8.6-algebra-2.pdfApr 08, 2014 · Changing log e x y= to exponential form would give e xy =. Evaluating log e](https://reader035.fdocuments.in/reader035/viewer/2022070911/5fb18f72f20cc62f47153453/html5/thumbnails/75.jpg)
Name _____________________________ Class ___________________Date ____________
Practice 8-6 Natural Logarithms Remember that common logarithms are logarithms of base 10. 4 4
10log3 log 3x x+ +=
e is the base of the Natural Logarithms, often abbreviated as ln. ( ) ( )log ln xe x =
Often called Euler’s number, e is an irrational that has a value of 2.718281828459045…
Changing loge x y= to exponential form would give ye x= .
Evaluating loge x There are two keys on the TI-84 that are for evaluating exponential functions with base e. They are the second function of the / (e) and L (ex) keys. Evaluate each expression to the nearest thousandth.
1. 5e ________________ 2. 4e− ________________ 3. 1
3e ________________ The same properties (product, quotient and power) of exponents apply to natural logarithms. So 33ln ln lnx y x y+ = Write as a single natural logarithm.
4. 4 ln 2 ln f− = ________________ 5. 1
lnx 3ln2
y+ = ________________
Solving natural logarithmic equations.
Solve ( )2ln 3 5 4x + =
Write in exponential form. ( )24 3 5e x= +
Take the square root of both sides. 4 3 5e x± = +
Subtract 5 from both sides. 4 5 3e x± − =
Divide both sides by 3. 4 5
3
ex
± − =
Evaluate using the calculator. 7.39 and 4.130x x= − =
6. Solve ln 9 5x = 7. Solve 2
ln 123
x + =
![Page 76: Practice 8-6 Natural Logarithmsstaff.bbhcsd.org/bradacm/files/2014/04/8.6-algebra-2.pdfApr 08, 2014 · Changing log e x y= to exponential form would give e xy =. Evaluating log e](https://reader035.fdocuments.in/reader035/viewer/2022070911/5fb18f72f20cc62f47153453/html5/thumbnails/76.jpg)
Solving an exponential equation. Solve 27 2.5 20xe + = Subtract 2.5 from both sides to start 27 17.5xe = isolating the exponential term. Divide both sides by 7 to finish 2 2.5xe = isolating the exponential term. Take the natural log of both sides. 2ln ln 2.5xe = Use the power property. 2 x ln ln 2.5e = Simplify using ln 1e = . 2 x ln 2.5=
Divide by 2. ln 2.5
x2
=
Use the calculator to finish. x 0.458≈ 8. Solve 1 30xe + = 9. Solve 34 1.2 14xe + =
Solve each equation. Round to the nearest ten-thousandth. Check your answer. Show all work for each. Circle your answer. 10. 18xe = 11. 2 12xe = 12. ln 3 6x =
13. ( )ln 4 1 36x − =
14. 5 4 7x
e + = 15. 22ln 2 x 1=
lne = 1
![Page 77: Practice 8-6 Natural Logarithmsstaff.bbhcsd.org/bradacm/files/2014/04/8.6-algebra-2.pdfApr 08, 2014 · Changing log e x y= to exponential form would give e xy =. Evaluating log e](https://reader035.fdocuments.in/reader035/viewer/2022070911/5fb18f72f20cc62f47153453/html5/thumbnails/77.jpg)
Name _____________________________ Class ___________________Date ____________
Practice 8-6 Natural Logarithms Remember that common logarithms are logarithms of base 10. 4 4
10log3 log 3x x+ +=
e is the base of the Natural Logarithms, often abbreviated as ln. ( ) ( )log ln xe x =
Often called Euler’s number, e is an irrational that has a value of 2.718281828459045…
Changing loge x y= to exponential form would give ye x= .
Evaluating loge x There are two keys on the TI-84 that are for evaluating exponential functions with base e. They are the second function of the / (e) and L (ex) keys. Evaluate each expression to the nearest thousandth.
1. 5e ________________ 2. 4e− ________________ 3. 1
3e ________________ The same properties (product, quotient and power) of exponents apply to natural logarithms. So 33ln ln lnx y x y+ = Write as a single natural logarithm.
4. 4 ln 2 ln f− = ________________ 5. 1
lnx 3ln2
y+ = ________________
Solving natural logarithmic equations.
Solve ( )2ln 3 5 4x + =
Write in exponential form. ( )24 3 5e x= +
Take the square root of both sides. 4 3 5e x± = +
Subtract 5 from both sides. 4 5 3e x± − =
Divide both sides by 3. 4 5
3
ex
± − =
Evaluate using the calculator. 7.39 and 4.130x x= − =
6. Solve ln 9 5x = 7. Solve 2
ln 123
x + =
![Page 78: Practice 8-6 Natural Logarithmsstaff.bbhcsd.org/bradacm/files/2014/04/8.6-algebra-2.pdfApr 08, 2014 · Changing log e x y= to exponential form would give e xy =. Evaluating log e](https://reader035.fdocuments.in/reader035/viewer/2022070911/5fb18f72f20cc62f47153453/html5/thumbnails/78.jpg)
Solving an exponential equation. Solve 27 2.5 20xe + = Subtract 2.5 from both sides to start 27 17.5xe = isolating the exponential term. Divide both sides by 7 to finish 2 2.5xe = isolating the exponential term. Take the natural log of both sides. 2ln ln 2.5xe = Use the power property. 2 x ln ln 2.5e = Simplify using ln 1e = . 2 x ln 2.5=
Divide by 2. ln 2.5
x2
=
Use the calculator to finish. x 0.458≈ 8. Solve 1 30xe + = 9. Solve 34 1.2 14xe + =
Solve each equation. Round to the nearest ten-thousandth. Check your answer. Show all work for each. Circle your answer. 10. 18xe = 11. 2 12xe = 12. ln 3 6x =
13. ( )ln 4 1 36x − =
14. 5 4 7x
e + = 15. 22ln 2 x 1=
lne = 1
![Page 79: Practice 8-6 Natural Logarithmsstaff.bbhcsd.org/bradacm/files/2014/04/8.6-algebra-2.pdfApr 08, 2014 · Changing log e x y= to exponential form would give e xy =. Evaluating log e](https://reader035.fdocuments.in/reader035/viewer/2022070911/5fb18f72f20cc62f47153453/html5/thumbnails/79.jpg)
Name _____________________________ Class ___________________Date ____________
Practice 8-6 Natural Logarithms Remember that common logarithms are logarithms of base 10. 4 4
10log3 log 3x x+ +=
e is the base of the Natural Logarithms, often abbreviated as ln. ( ) ( )log ln xe x =
Often called Euler’s number, e is an irrational that has a value of 2.718281828459045…
Changing loge x y= to exponential form would give ye x= .
Evaluating loge x There are two keys on the TI-84 that are for evaluating exponential functions with base e. They are the second function of the / (e) and L (ex) keys. Evaluate each expression to the nearest thousandth.
1. 5e ________________ 2. 4e− ________________ 3. 1
3e ________________ The same properties (product, quotient and power) of exponents apply to natural logarithms. So 33ln ln lnx y x y+ = Write as a single natural logarithm.
4. 4 ln 2 ln f− = ________________ 5. 1
lnx 3ln2
y+ = ________________
Solving natural logarithmic equations.
Solve ( )2ln 3 5 4x + =
Write in exponential form. ( )24 3 5e x= +
Take the square root of both sides. 4 3 5e x± = +
Subtract 5 from both sides. 4 5 3e x± − =
Divide both sides by 3. 4 5
3
ex
± − =
Evaluate using the calculator. 7.39 and 4.130x x= − =
6. Solve ln 9 5x = 7. Solve 2
ln 123
x + =
![Page 80: Practice 8-6 Natural Logarithmsstaff.bbhcsd.org/bradacm/files/2014/04/8.6-algebra-2.pdfApr 08, 2014 · Changing log e x y= to exponential form would give e xy =. Evaluating log e](https://reader035.fdocuments.in/reader035/viewer/2022070911/5fb18f72f20cc62f47153453/html5/thumbnails/80.jpg)
Solving an exponential equation. Solve 27 2.5 20xe + = Subtract 2.5 from both sides to start 27 17.5xe = isolating the exponential term. Divide both sides by 7 to finish 2 2.5xe = isolating the exponential term. Take the natural log of both sides. 2ln ln 2.5xe = Use the power property. 2 x ln ln 2.5e = Simplify using ln 1e = . 2 x ln 2.5=
Divide by 2. ln 2.5
x2
=
Use the calculator to finish. x 0.458≈ 8. Solve 1 30xe + = 9. Solve 34 1.2 14xe + =
Solve each equation. Round to the nearest ten-thousandth. Check your answer. Show all work for each. Circle your answer. 10. 18xe = 11. 2 12xe = 12. ln 3 6x =
13. ( )ln 4 1 36x − =
14. 5 4 7x
e + = 15. 22ln 2 x 1=
lne = 1
![Page 81: Practice 8-6 Natural Logarithmsstaff.bbhcsd.org/bradacm/files/2014/04/8.6-algebra-2.pdfApr 08, 2014 · Changing log e x y= to exponential form would give e xy =. Evaluating log e](https://reader035.fdocuments.in/reader035/viewer/2022070911/5fb18f72f20cc62f47153453/html5/thumbnails/81.jpg)
Name _____________________________ Class ___________________Date ____________
Practice 8-6 Natural Logarithms Remember that common logarithms are logarithms of base 10. 4 4
10log3 log 3x x+ +=
e is the base of the Natural Logarithms, often abbreviated as ln. ( ) ( )log ln xe x =
Often called Euler’s number, e is an irrational that has a value of 2.718281828459045…
Changing loge x y= to exponential form would give ye x= .
Evaluating loge x There are two keys on the TI-84 that are for evaluating exponential functions with base e. They are the second function of the / (e) and L (ex) keys. Evaluate each expression to the nearest thousandth.
1. 5e ________________ 2. 4e− ________________ 3. 1
3e ________________ The same properties (product, quotient and power) of exponents apply to natural logarithms. So 33ln ln lnx y x y+ = Write as a single natural logarithm.
4. 4 ln 2 ln f− = ________________ 5. 1
lnx 3ln2
y+ = ________________
Solving natural logarithmic equations.
Solve ( )2ln 3 5 4x + =
Write in exponential form. ( )24 3 5e x= +
Take the square root of both sides. 4 3 5e x± = +
Subtract 5 from both sides. 4 5 3e x± − =
Divide both sides by 3. 4 5
3
ex
± − =
Evaluate using the calculator. 7.39 and 4.130x x= − =
6. Solve ln 9 5x = 7. Solve 2
ln 123
x + =
![Page 82: Practice 8-6 Natural Logarithmsstaff.bbhcsd.org/bradacm/files/2014/04/8.6-algebra-2.pdfApr 08, 2014 · Changing log e x y= to exponential form would give e xy =. Evaluating log e](https://reader035.fdocuments.in/reader035/viewer/2022070911/5fb18f72f20cc62f47153453/html5/thumbnails/82.jpg)
Solving an exponential equation. Solve 27 2.5 20xe + = Subtract 2.5 from both sides to start 27 17.5xe = isolating the exponential term. Divide both sides by 7 to finish 2 2.5xe = isolating the exponential term. Take the natural log of both sides. 2ln ln 2.5xe = Use the power property. 2 x ln ln 2.5e = Simplify using ln 1e = . 2 x ln 2.5=
Divide by 2. ln 2.5
x2
=
Use the calculator to finish. x 0.458≈ 8. Solve 1 30xe + = 9. Solve 34 1.2 14xe + =
Solve each equation. Round to the nearest ten-thousandth. Check your answer. Show all work for each. Circle your answer. 10. 18xe = 11. 2 12xe = 12. ln 3 6x =
13. ( )ln 4 1 36x − =
14. 5 4 7x
e + = 15. 22ln 2 x 1=
lne = 1
![Page 83: Practice 8-6 Natural Logarithmsstaff.bbhcsd.org/bradacm/files/2014/04/8.6-algebra-2.pdfApr 08, 2014 · Changing log e x y= to exponential form would give e xy =. Evaluating log e](https://reader035.fdocuments.in/reader035/viewer/2022070911/5fb18f72f20cc62f47153453/html5/thumbnails/83.jpg)
Name _____________________________ Class ___________________Date ____________
Practice 8-6 Natural Logarithms Remember that common logarithms are logarithms of base 10. 4 4
10log3 log 3x x+ +=
e is the base of the Natural Logarithms, often abbreviated as ln. ( ) ( )log ln xe x =
Often called Euler’s number, e is an irrational that has a value of 2.718281828459045…
Changing loge x y= to exponential form would give ye x= .
Evaluating loge x There are two keys on the TI-84 that are for evaluating exponential functions with base e. They are the second function of the / (e) and L (ex) keys. Evaluate each expression to the nearest thousandth.
1. 5e ________________ 2. 4e− ________________ 3. 1
3e ________________ The same properties (product, quotient and power) of exponents apply to natural logarithms. So 33ln ln lnx y x y+ = Write as a single natural logarithm.
4. 4 ln 2 ln f− = ________________ 5. 1
lnx 3ln2
y+ = ________________
Solving natural logarithmic equations.
Solve ( )2ln 3 5 4x + =
Write in exponential form. ( )24 3 5e x= +
Take the square root of both sides. 4 3 5e x± = +
Subtract 5 from both sides. 4 5 3e x± − =
Divide both sides by 3. 4 5
3
ex
± − =
Evaluate using the calculator. 7.39 and 4.130x x= − =
6. Solve ln 9 5x = 7. Solve 2
ln 123
x + =
![Page 84: Practice 8-6 Natural Logarithmsstaff.bbhcsd.org/bradacm/files/2014/04/8.6-algebra-2.pdfApr 08, 2014 · Changing log e x y= to exponential form would give e xy =. Evaluating log e](https://reader035.fdocuments.in/reader035/viewer/2022070911/5fb18f72f20cc62f47153453/html5/thumbnails/84.jpg)
Solving an exponential equation. Solve 27 2.5 20xe + = Subtract 2.5 from both sides to start 27 17.5xe = isolating the exponential term. Divide both sides by 7 to finish 2 2.5xe = isolating the exponential term. Take the natural log of both sides. 2ln ln 2.5xe = Use the power property. 2 x ln ln 2.5e = Simplify using ln 1e = . 2 x ln 2.5=
Divide by 2. ln 2.5
x2
=
Use the calculator to finish. x 0.458≈ 8. Solve 1 30xe + = 9. Solve 34 1.2 14xe + =
Solve each equation. Round to the nearest ten-thousandth. Check your answer. Show all work for each. Circle your answer. 10. 18xe = 11. 2 12xe = 12. ln 3 6x =
13. ( )ln 4 1 36x − =
14. 5 4 7x
e + = 15. 22ln 2 x 1=
lne = 1
![Page 85: Practice 8-6 Natural Logarithmsstaff.bbhcsd.org/bradacm/files/2014/04/8.6-algebra-2.pdfApr 08, 2014 · Changing log e x y= to exponential form would give e xy =. Evaluating log e](https://reader035.fdocuments.in/reader035/viewer/2022070911/5fb18f72f20cc62f47153453/html5/thumbnails/85.jpg)
Name _____________________________ Class ___________________Date ____________
Practice 8-6 Natural Logarithms Remember that common logarithms are logarithms of base 10. 4 4
10log3 log 3x x+ +=
e is the base of the Natural Logarithms, often abbreviated as ln. ( ) ( )log ln xe x =
Often called Euler’s number, e is an irrational that has a value of 2.718281828459045…
Changing loge x y= to exponential form would give ye x= .
Evaluating loge x There are two keys on the TI-84 that are for evaluating exponential functions with base e. They are the second function of the / (e) and L (ex) keys. Evaluate each expression to the nearest thousandth.
1. 5e ________________ 2. 4e− ________________ 3. 1
3e ________________ The same properties (product, quotient and power) of exponents apply to natural logarithms. So 33ln ln lnx y x y+ = Write as a single natural logarithm.
4. 4 ln 2 ln f− = ________________ 5. 1
lnx 3ln2
y+ = ________________
Solving natural logarithmic equations.
Solve ( )2ln 3 5 4x + =
Write in exponential form. ( )24 3 5e x= +
Take the square root of both sides. 4 3 5e x± = +
Subtract 5 from both sides. 4 5 3e x± − =
Divide both sides by 3. 4 5
3
ex
± − =
Evaluate using the calculator. 7.39 and 4.130x x= − =
6. Solve ln 9 5x = 7. Solve 2
ln 123
x + =
![Page 86: Practice 8-6 Natural Logarithmsstaff.bbhcsd.org/bradacm/files/2014/04/8.6-algebra-2.pdfApr 08, 2014 · Changing log e x y= to exponential form would give e xy =. Evaluating log e](https://reader035.fdocuments.in/reader035/viewer/2022070911/5fb18f72f20cc62f47153453/html5/thumbnails/86.jpg)
Solving an exponential equation. Solve 27 2.5 20xe + = Subtract 2.5 from both sides to start 27 17.5xe = isolating the exponential term. Divide both sides by 7 to finish 2 2.5xe = isolating the exponential term. Take the natural log of both sides. 2ln ln 2.5xe = Use the power property. 2 x ln ln 2.5e = Simplify using ln 1e = . 2 x ln 2.5=
Divide by 2. ln 2.5
x2
=
Use the calculator to finish. x 0.458≈ 8. Solve 1 30xe + = 9. Solve 34 1.2 14xe + =
Solve each equation. Round to the nearest ten-thousandth. Check your answer. Show all work for each. Circle your answer. 10. 18xe = 11. 2 12xe = 12. ln 3 6x =
13. ( )ln 4 1 36x − =
14. 5 4 7x
e + = 15. 22ln 2 x 1=
lne = 1
![Page 87: Practice 8-6 Natural Logarithmsstaff.bbhcsd.org/bradacm/files/2014/04/8.6-algebra-2.pdfApr 08, 2014 · Changing log e x y= to exponential form would give e xy =. Evaluating log e](https://reader035.fdocuments.in/reader035/viewer/2022070911/5fb18f72f20cc62f47153453/html5/thumbnails/87.jpg)
Name _____________________________ Class ___________________Date ____________
Practice 8-6 Natural Logarithms Remember that common logarithms are logarithms of base 10. 4 4
10log3 log 3x x+ +=
e is the base of the Natural Logarithms, often abbreviated as ln. ( ) ( )log ln xe x =
Often called Euler’s number, e is an irrational that has a value of 2.718281828459045…
Changing loge x y= to exponential form would give ye x= .
Evaluating loge x There are two keys on the TI-84 that are for evaluating exponential functions with base e. They are the second function of the / (e) and L (ex) keys. Evaluate each expression to the nearest thousandth.
1. 5e ________________ 2. 4e− ________________ 3. 1
3e ________________ The same properties (product, quotient and power) of exponents apply to natural logarithms. So 33ln ln lnx y x y+ = Write as a single natural logarithm.
4. 4 ln 2 ln f− = ________________ 5. 1
lnx 3ln2
y+ = ________________
Solving natural logarithmic equations.
Solve ( )2ln 3 5 4x + =
Write in exponential form. ( )24 3 5e x= +
Take the square root of both sides. 4 3 5e x± = +
Subtract 5 from both sides. 4 5 3e x± − =
Divide both sides by 3. 4 5
3
ex
± − =
Evaluate using the calculator. 7.39 and 4.130x x= − =
6. Solve ln 9 5x = 7. Solve 2
ln 123
x + =
![Page 88: Practice 8-6 Natural Logarithmsstaff.bbhcsd.org/bradacm/files/2014/04/8.6-algebra-2.pdfApr 08, 2014 · Changing log e x y= to exponential form would give e xy =. Evaluating log e](https://reader035.fdocuments.in/reader035/viewer/2022070911/5fb18f72f20cc62f47153453/html5/thumbnails/88.jpg)
Solving an exponential equation. Solve 27 2.5 20xe + = Subtract 2.5 from both sides to start 27 17.5xe = isolating the exponential term. Divide both sides by 7 to finish 2 2.5xe = isolating the exponential term. Take the natural log of both sides. 2ln ln 2.5xe = Use the power property. 2 x ln ln 2.5e = Simplify using ln 1e = . 2 x ln 2.5=
Divide by 2. ln 2.5
x2
=
Use the calculator to finish. x 0.458≈ 8. Solve 1 30xe + = 9. Solve 34 1.2 14xe + =
Solve each equation. Round to the nearest ten-thousandth. Check your answer. Show all work for each. Circle your answer. 10. 18xe = 11. 2 12xe = 12. ln 3 6x =
13. ( )ln 4 1 36x − =
14. 5 4 7x
e + = 15. 22ln 2 x 1=
lne = 1
![Page 89: Practice 8-6 Natural Logarithmsstaff.bbhcsd.org/bradacm/files/2014/04/8.6-algebra-2.pdfApr 08, 2014 · Changing log e x y= to exponential form would give e xy =. Evaluating log e](https://reader035.fdocuments.in/reader035/viewer/2022070911/5fb18f72f20cc62f47153453/html5/thumbnails/89.jpg)
Name _____________________________ Class ___________________Date ____________
Practice 8-6 Natural Logarithms Remember that common logarithms are logarithms of base 10. 4 4
10log3 log 3x x+ +=
e is the base of the Natural Logarithms, often abbreviated as ln. ( ) ( )log ln xe x =
Often called Euler’s number, e is an irrational that has a value of 2.718281828459045…
Changing loge x y= to exponential form would give ye x= .
Evaluating loge x There are two keys on the TI-84 that are for evaluating exponential functions with base e. They are the second function of the / (e) and L (ex) keys. Evaluate each expression to the nearest thousandth.
1. 5e ________________ 2. 4e− ________________ 3. 1
3e ________________ The same properties (product, quotient and power) of exponents apply to natural logarithms. So 33ln ln lnx y x y+ = Write as a single natural logarithm.
4. 4 ln 2 ln f− = ________________ 5. 1
lnx 3ln2
y+ = ________________
Solving natural logarithmic equations.
Solve ( )2ln 3 5 4x + =
Write in exponential form. ( )24 3 5e x= +
Take the square root of both sides. 4 3 5e x± = +
Subtract 5 from both sides. 4 5 3e x± − =
Divide both sides by 3. 4 5
3
ex
± − =
Evaluate using the calculator. 7.39 and 4.130x x= − =
6. Solve ln 9 5x = 7. Solve 2
ln 123
x + =
![Page 90: Practice 8-6 Natural Logarithmsstaff.bbhcsd.org/bradacm/files/2014/04/8.6-algebra-2.pdfApr 08, 2014 · Changing log e x y= to exponential form would give e xy =. Evaluating log e](https://reader035.fdocuments.in/reader035/viewer/2022070911/5fb18f72f20cc62f47153453/html5/thumbnails/90.jpg)
Solving an exponential equation. Solve 27 2.5 20xe + = Subtract 2.5 from both sides to start 27 17.5xe = isolating the exponential term. Divide both sides by 7 to finish 2 2.5xe = isolating the exponential term. Take the natural log of both sides. 2ln ln 2.5xe = Use the power property. 2 x ln ln 2.5e = Simplify using ln 1e = . 2 x ln 2.5=
Divide by 2. ln 2.5
x2
=
Use the calculator to finish. x 0.458≈ 8. Solve 1 30xe + = 9. Solve 34 1.2 14xe + =
Solve each equation. Round to the nearest ten-thousandth. Check your answer. Show all work for each. Circle your answer. 10. 18xe = 11. 2 12xe = 12. ln 3 6x =
13. ( )ln 4 1 36x − =
14. 5 4 7x
e + = 15. 22ln 2 x 1=
lne = 1
![Page 91: Practice 8-6 Natural Logarithmsstaff.bbhcsd.org/bradacm/files/2014/04/8.6-algebra-2.pdfApr 08, 2014 · Changing log e x y= to exponential form would give e xy =. Evaluating log e](https://reader035.fdocuments.in/reader035/viewer/2022070911/5fb18f72f20cc62f47153453/html5/thumbnails/91.jpg)
Name _____________________________ Class ___________________Date ____________
Practice 8-6 Natural Logarithms Remember that common logarithms are logarithms of base 10. 4 4
10log3 log 3x x+ +=
e is the base of the Natural Logarithms, often abbreviated as ln. ( ) ( )log ln xe x =
Often called Euler’s number, e is an irrational that has a value of 2.718281828459045…
Changing loge x y= to exponential form would give ye x= .
Evaluating loge x There are two keys on the TI-84 that are for evaluating exponential functions with base e. They are the second function of the / (e) and L (ex) keys. Evaluate each expression to the nearest thousandth.
1. 5e ________________ 2. 4e− ________________ 3. 1
3e ________________ The same properties (product, quotient and power) of exponents apply to natural logarithms. So 33ln ln lnx y x y+ = Write as a single natural logarithm.
4. 4 ln 2 ln f− = ________________ 5. 1
lnx 3ln2
y+ = ________________
Solving natural logarithmic equations.
Solve ( )2ln 3 5 4x + =
Write in exponential form. ( )24 3 5e x= +
Take the square root of both sides. 4 3 5e x± = +
Subtract 5 from both sides. 4 5 3e x± − =
Divide both sides by 3. 4 5
3
ex
± − =
Evaluate using the calculator. 7.39 and 4.130x x= − =
6. Solve ln 9 5x = 7. Solve 2
ln 123
x + =
![Page 92: Practice 8-6 Natural Logarithmsstaff.bbhcsd.org/bradacm/files/2014/04/8.6-algebra-2.pdfApr 08, 2014 · Changing log e x y= to exponential form would give e xy =. Evaluating log e](https://reader035.fdocuments.in/reader035/viewer/2022070911/5fb18f72f20cc62f47153453/html5/thumbnails/92.jpg)
Solving an exponential equation. Solve 27 2.5 20xe + = Subtract 2.5 from both sides to start 27 17.5xe = isolating the exponential term. Divide both sides by 7 to finish 2 2.5xe = isolating the exponential term. Take the natural log of both sides. 2ln ln 2.5xe = Use the power property. 2 x ln ln 2.5e = Simplify using ln 1e = . 2 x ln 2.5=
Divide by 2. ln 2.5
x2
=
Use the calculator to finish. x 0.458≈ 8. Solve 1 30xe + = 9. Solve 34 1.2 14xe + =
Solve each equation. Round to the nearest ten-thousandth. Check your answer. Show all work for each. Circle your answer. 10. 18xe = 11. 2 12xe = 12. ln 3 6x =
13. ( )ln 4 1 36x − =
14. 5 4 7x
e + = 15. 22ln 2 x 1=
lne = 1
![Page 93: Practice 8-6 Natural Logarithmsstaff.bbhcsd.org/bradacm/files/2014/04/8.6-algebra-2.pdfApr 08, 2014 · Changing log e x y= to exponential form would give e xy =. Evaluating log e](https://reader035.fdocuments.in/reader035/viewer/2022070911/5fb18f72f20cc62f47153453/html5/thumbnails/93.jpg)
Name _____________________________ Class ___________________Date ____________
Practice 8-6 Natural Logarithms Remember that common logarithms are logarithms of base 10. 4 4
10log3 log 3x x+ +=
e is the base of the Natural Logarithms, often abbreviated as ln. ( ) ( )log ln xe x =
Often called Euler’s number, e is an irrational that has a value of 2.718281828459045…
Changing loge x y= to exponential form would give ye x= .
Evaluating loge x There are two keys on the TI-84 that are for evaluating exponential functions with base e. They are the second function of the / (e) and L (ex) keys. Evaluate each expression to the nearest thousandth.
1. 5e ________________ 2. 4e− ________________ 3. 1
3e ________________ The same properties (product, quotient and power) of exponents apply to natural logarithms. So 33ln ln lnx y x y+ = Write as a single natural logarithm.
4. 4 ln 2 ln f− = ________________ 5. 1
lnx 3ln2
y+ = ________________
Solving natural logarithmic equations.
Solve ( )2ln 3 5 4x + =
Write in exponential form. ( )24 3 5e x= +
Take the square root of both sides. 4 3 5e x± = +
Subtract 5 from both sides. 4 5 3e x± − =
Divide both sides by 3. 4 5
3
ex
± − =
Evaluate using the calculator. 7.39 and 4.130x x= − =
6. Solve ln 9 5x = 7. Solve 2
ln 123
x + =
![Page 94: Practice 8-6 Natural Logarithmsstaff.bbhcsd.org/bradacm/files/2014/04/8.6-algebra-2.pdfApr 08, 2014 · Changing log e x y= to exponential form would give e xy =. Evaluating log e](https://reader035.fdocuments.in/reader035/viewer/2022070911/5fb18f72f20cc62f47153453/html5/thumbnails/94.jpg)
Solving an exponential equation. Solve 27 2.5 20xe + = Subtract 2.5 from both sides to start 27 17.5xe = isolating the exponential term. Divide both sides by 7 to finish 2 2.5xe = isolating the exponential term. Take the natural log of both sides. 2ln ln 2.5xe = Use the power property. 2 x ln ln 2.5e = Simplify using ln 1e = . 2 x ln 2.5=
Divide by 2. ln 2.5
x2
=
Use the calculator to finish. x 0.458≈ 8. Solve 1 30xe + = 9. Solve 34 1.2 14xe + =
Solve each equation. Round to the nearest ten-thousandth. Check your answer. Show all work for each. Circle your answer. 10. 18xe = 11. 2 12xe = 12. ln 3 6x =
13. ( )ln 4 1 36x − =
14. 5 4 7x
e + = 15. 22ln 2 x 1=
lne = 1
![Page 95: Practice 8-6 Natural Logarithmsstaff.bbhcsd.org/bradacm/files/2014/04/8.6-algebra-2.pdfApr 08, 2014 · Changing log e x y= to exponential form would give e xy =. Evaluating log e](https://reader035.fdocuments.in/reader035/viewer/2022070911/5fb18f72f20cc62f47153453/html5/thumbnails/95.jpg)
Name _____________________________ Class ___________________Date ____________
Practice 8-6 Natural Logarithms Remember that common logarithms are logarithms of base 10. 4 4
10log3 log 3x x+ +=
e is the base of the Natural Logarithms, often abbreviated as ln. ( ) ( )log ln xe x =
Often called Euler’s number, e is an irrational that has a value of 2.718281828459045…
Changing loge x y= to exponential form would give ye x= .
Evaluating loge x There are two keys on the TI-84 that are for evaluating exponential functions with base e. They are the second function of the / (e) and L (ex) keys. Evaluate each expression to the nearest thousandth.
1. 5e ________________ 2. 4e− ________________ 3. 1
3e ________________ The same properties (product, quotient and power) of exponents apply to natural logarithms. So 33ln ln lnx y x y+ = Write as a single natural logarithm.
4. 4 ln 2 ln f− = ________________ 5. 1
lnx 3ln2
y+ = ________________
Solving natural logarithmic equations.
Solve ( )2ln 3 5 4x + =
Write in exponential form. ( )24 3 5e x= +
Take the square root of both sides. 4 3 5e x± = +
Subtract 5 from both sides. 4 5 3e x± − =
Divide both sides by 3. 4 5
3
ex
± − =
Evaluate using the calculator. 7.39 and 4.130x x= − =
6. Solve ln 9 5x = 7. Solve 2
ln 123
x + =
![Page 96: Practice 8-6 Natural Logarithmsstaff.bbhcsd.org/bradacm/files/2014/04/8.6-algebra-2.pdfApr 08, 2014 · Changing log e x y= to exponential form would give e xy =. Evaluating log e](https://reader035.fdocuments.in/reader035/viewer/2022070911/5fb18f72f20cc62f47153453/html5/thumbnails/96.jpg)
Solving an exponential equation. Solve 27 2.5 20xe + = Subtract 2.5 from both sides to start 27 17.5xe = isolating the exponential term. Divide both sides by 7 to finish 2 2.5xe = isolating the exponential term. Take the natural log of both sides. 2ln ln 2.5xe = Use the power property. 2 x ln ln 2.5e = Simplify using ln 1e = . 2 x ln 2.5=
Divide by 2. ln 2.5
x2
=
Use the calculator to finish. x 0.458≈ 8. Solve 1 30xe + = 9. Solve 34 1.2 14xe + =
Solve each equation. Round to the nearest ten-thousandth. Check your answer. Show all work for each. Circle your answer. 10. 18xe = 11. 2 12xe = 12. ln 3 6x =
13. ( )ln 4 1 36x − =
14. 5 4 7x
e + = 15. 22ln 2 x 1=
lne = 1
![Page 97: Practice 8-6 Natural Logarithmsstaff.bbhcsd.org/bradacm/files/2014/04/8.6-algebra-2.pdfApr 08, 2014 · Changing log e x y= to exponential form would give e xy =. Evaluating log e](https://reader035.fdocuments.in/reader035/viewer/2022070911/5fb18f72f20cc62f47153453/html5/thumbnails/97.jpg)
Name _____________________________ Class ___________________Date ____________
Practice 8-6 Natural Logarithms Remember that common logarithms are logarithms of base 10. 4 4
10log3 log 3x x+ +=
e is the base of the Natural Logarithms, often abbreviated as ln. ( ) ( )log ln xe x =
Often called Euler’s number, e is an irrational that has a value of 2.718281828459045…
Changing loge x y= to exponential form would give ye x= .
Evaluating loge x There are two keys on the TI-84 that are for evaluating exponential functions with base e. They are the second function of the / (e) and L (ex) keys. Evaluate each expression to the nearest thousandth.
1. 5e ________________ 2. 4e− ________________ 3. 1
3e ________________ The same properties (product, quotient and power) of exponents apply to natural logarithms. So 33ln ln lnx y x y+ = Write as a single natural logarithm.
4. 4 ln 2 ln f− = ________________ 5. 1
lnx 3ln2
y+ = ________________
Solving natural logarithmic equations.
Solve ( )2ln 3 5 4x + =
Write in exponential form. ( )24 3 5e x= +
Take the square root of both sides. 4 3 5e x± = +
Subtract 5 from both sides. 4 5 3e x± − =
Divide both sides by 3. 4 5
3
ex
± − =
Evaluate using the calculator. 7.39 and 4.130x x= − =
6. Solve ln 9 5x = 7. Solve 2
ln 123
x + =
![Page 98: Practice 8-6 Natural Logarithmsstaff.bbhcsd.org/bradacm/files/2014/04/8.6-algebra-2.pdfApr 08, 2014 · Changing log e x y= to exponential form would give e xy =. Evaluating log e](https://reader035.fdocuments.in/reader035/viewer/2022070911/5fb18f72f20cc62f47153453/html5/thumbnails/98.jpg)
Solving an exponential equation. Solve 27 2.5 20xe + = Subtract 2.5 from both sides to start 27 17.5xe = isolating the exponential term. Divide both sides by 7 to finish 2 2.5xe = isolating the exponential term. Take the natural log of both sides. 2ln ln 2.5xe = Use the power property. 2 x ln ln 2.5e = Simplify using ln 1e = . 2 x ln 2.5=
Divide by 2. ln 2.5
x2
=
Use the calculator to finish. x 0.458≈ 8. Solve 1 30xe + = 9. Solve 34 1.2 14xe + =
Solve each equation. Round to the nearest ten-thousandth. Check your answer. Show all work for each. Circle your answer. 10. 18xe = 11. 2 12xe = 12. ln 3 6x =
13. ( )ln 4 1 36x − =
14. 5 4 7x
e + = 15. 22ln 2 x 1=
lne = 1
![Page 99: Practice 8-6 Natural Logarithmsstaff.bbhcsd.org/bradacm/files/2014/04/8.6-algebra-2.pdfApr 08, 2014 · Changing log e x y= to exponential form would give e xy =. Evaluating log e](https://reader035.fdocuments.in/reader035/viewer/2022070911/5fb18f72f20cc62f47153453/html5/thumbnails/99.jpg)
Name _____________________________ Class ___________________Date ____________
Practice 8-6 Natural Logarithms Remember that common logarithms are logarithms of base 10. 4 4
10log3 log 3x x+ +=
e is the base of the Natural Logarithms, often abbreviated as ln. ( ) ( )log ln xe x =
Often called Euler’s number, e is an irrational that has a value of 2.718281828459045…
Changing loge x y= to exponential form would give ye x= .
Evaluating loge x There are two keys on the TI-84 that are for evaluating exponential functions with base e. They are the second function of the / (e) and L (ex) keys. Evaluate each expression to the nearest thousandth.
1. 5e ________________ 2. 4e− ________________ 3. 1
3e ________________ The same properties (product, quotient and power) of exponents apply to natural logarithms. So 33ln ln lnx y x y+ = Write as a single natural logarithm.
4. 4 ln 2 ln f− = ________________ 5. 1
lnx 3ln2
y+ = ________________
Solving natural logarithmic equations.
Solve ( )2ln 3 5 4x + =
Write in exponential form. ( )24 3 5e x= +
Take the square root of both sides. 4 3 5e x± = +
Subtract 5 from both sides. 4 5 3e x± − =
Divide both sides by 3. 4 5
3
ex
± − =
Evaluate using the calculator. 7.39 and 4.130x x= − =
6. Solve ln 9 5x = 7. Solve 2
ln 123
x + =
![Page 100: Practice 8-6 Natural Logarithmsstaff.bbhcsd.org/bradacm/files/2014/04/8.6-algebra-2.pdfApr 08, 2014 · Changing log e x y= to exponential form would give e xy =. Evaluating log e](https://reader035.fdocuments.in/reader035/viewer/2022070911/5fb18f72f20cc62f47153453/html5/thumbnails/100.jpg)
Solving an exponential equation. Solve 27 2.5 20xe + = Subtract 2.5 from both sides to start 27 17.5xe = isolating the exponential term. Divide both sides by 7 to finish 2 2.5xe = isolating the exponential term. Take the natural log of both sides. 2ln ln 2.5xe = Use the power property. 2 x ln ln 2.5e = Simplify using ln 1e = . 2 x ln 2.5=
Divide by 2. ln 2.5
x2
=
Use the calculator to finish. x 0.458≈ 8. Solve 1 30xe + = 9. Solve 34 1.2 14xe + =
Solve each equation. Round to the nearest ten-thousandth. Check your answer. Show all work for each. Circle your answer. 10. 18xe = 11. 2 12xe = 12. ln 3 6x =
13. ( )ln 4 1 36x − =
14. 5 4 7x
e + = 15. 22ln 2 x 1=
lne = 1
![Page 101: Practice 8-6 Natural Logarithmsstaff.bbhcsd.org/bradacm/files/2014/04/8.6-algebra-2.pdfApr 08, 2014 · Changing log e x y= to exponential form would give e xy =. Evaluating log e](https://reader035.fdocuments.in/reader035/viewer/2022070911/5fb18f72f20cc62f47153453/html5/thumbnails/101.jpg)
Name _____________________________ Class ___________________Date ____________
Practice 8-6 Natural Logarithms Remember that common logarithms are logarithms of base 10. 4 4
10log3 log 3x x+ +=
e is the base of the Natural Logarithms, often abbreviated as ln. ( ) ( )log ln xe x =
Often called Euler’s number, e is an irrational that has a value of 2.718281828459045…
Changing loge x y= to exponential form would give ye x= .
Evaluating loge x There are two keys on the TI-84 that are for evaluating exponential functions with base e. They are the second function of the / (e) and L (ex) keys. Evaluate each expression to the nearest thousandth.
1. 5e ________________ 2. 4e− ________________ 3. 1
3e ________________ The same properties (product, quotient and power) of exponents apply to natural logarithms. So 33ln ln lnx y x y+ = Write as a single natural logarithm.
4. 4 ln 2 ln f− = ________________ 5. 1
lnx 3ln2
y+ = ________________
Solving natural logarithmic equations.
Solve ( )2ln 3 5 4x + =
Write in exponential form. ( )24 3 5e x= +
Take the square root of both sides. 4 3 5e x± = +
Subtract 5 from both sides. 4 5 3e x± − =
Divide both sides by 3. 4 5
3
ex
± − =
Evaluate using the calculator. 7.39 and 4.130x x= − =
6. Solve ln 9 5x = 7. Solve 2
ln 123
x + =
![Page 102: Practice 8-6 Natural Logarithmsstaff.bbhcsd.org/bradacm/files/2014/04/8.6-algebra-2.pdfApr 08, 2014 · Changing log e x y= to exponential form would give e xy =. Evaluating log e](https://reader035.fdocuments.in/reader035/viewer/2022070911/5fb18f72f20cc62f47153453/html5/thumbnails/102.jpg)
Solving an exponential equation. Solve 27 2.5 20xe + = Subtract 2.5 from both sides to start 27 17.5xe = isolating the exponential term. Divide both sides by 7 to finish 2 2.5xe = isolating the exponential term. Take the natural log of both sides. 2ln ln 2.5xe = Use the power property. 2 x ln ln 2.5e = Simplify using ln 1e = . 2 x ln 2.5=
Divide by 2. ln 2.5
x2
=
Use the calculator to finish. x 0.458≈ 8. Solve 1 30xe + = 9. Solve 34 1.2 14xe + =
Solve each equation. Round to the nearest ten-thousandth. Check your answer. Show all work for each. Circle your answer. 10. 18xe = 11. 2 12xe = 12. ln 3 6x =
13. ( )ln 4 1 36x − =
14. 5 4 7x
e + = 15. 22ln 2 x 1=
lne = 1
![Page 103: Practice 8-6 Natural Logarithmsstaff.bbhcsd.org/bradacm/files/2014/04/8.6-algebra-2.pdfApr 08, 2014 · Changing log e x y= to exponential form would give e xy =. Evaluating log e](https://reader035.fdocuments.in/reader035/viewer/2022070911/5fb18f72f20cc62f47153453/html5/thumbnails/103.jpg)
Name _____________________________ Class ___________________Date ____________
Practice 8-6 Natural Logarithms Remember that common logarithms are logarithms of base 10. 4 4
10log3 log 3x x+ +=
e is the base of the Natural Logarithms, often abbreviated as ln. ( ) ( )log ln xe x =
Often called Euler’s number, e is an irrational that has a value of 2.718281828459045…
Changing loge x y= to exponential form would give ye x= .
Evaluating loge x There are two keys on the TI-84 that are for evaluating exponential functions with base e. They are the second function of the / (e) and L (ex) keys. Evaluate each expression to the nearest thousandth.
1. 5e ________________ 2. 4e− ________________ 3. 1
3e ________________ The same properties (product, quotient and power) of exponents apply to natural logarithms. So 33ln ln lnx y x y+ = Write as a single natural logarithm.
4. 4 ln 2 ln f− = ________________ 5. 1
lnx 3ln2
y+ = ________________
Solving natural logarithmic equations.
Solve ( )2ln 3 5 4x + =
Write in exponential form. ( )24 3 5e x= +
Take the square root of both sides. 4 3 5e x± = +
Subtract 5 from both sides. 4 5 3e x± − =
Divide both sides by 3. 4 5
3
ex
± − =
Evaluate using the calculator. 7.39 and 4.130x x= − =
6. Solve ln 9 5x = 7. Solve 2
ln 123
x + =
![Page 104: Practice 8-6 Natural Logarithmsstaff.bbhcsd.org/bradacm/files/2014/04/8.6-algebra-2.pdfApr 08, 2014 · Changing log e x y= to exponential form would give e xy =. Evaluating log e](https://reader035.fdocuments.in/reader035/viewer/2022070911/5fb18f72f20cc62f47153453/html5/thumbnails/104.jpg)
Solving an exponential equation. Solve 27 2.5 20xe + = Subtract 2.5 from both sides to start 27 17.5xe = isolating the exponential term. Divide both sides by 7 to finish 2 2.5xe = isolating the exponential term. Take the natural log of both sides. 2ln ln 2.5xe = Use the power property. 2 x ln ln 2.5e = Simplify using ln 1e = . 2 x ln 2.5=
Divide by 2. ln 2.5
x2
=
Use the calculator to finish. x 0.458≈ 8. Solve 1 30xe + = 9. Solve 34 1.2 14xe + =
Solve each equation. Round to the nearest ten-thousandth. Check your answer. Show all work for each. Circle your answer. 10. 18xe = 11. 2 12xe = 12. ln 3 6x =
13. ( )ln 4 1 36x − =
14. 5 4 7x
e + = 15. 22ln 2 x 1=
lne = 1
![Page 105: Practice 8-6 Natural Logarithmsstaff.bbhcsd.org/bradacm/files/2014/04/8.6-algebra-2.pdfApr 08, 2014 · Changing log e x y= to exponential form would give e xy =. Evaluating log e](https://reader035.fdocuments.in/reader035/viewer/2022070911/5fb18f72f20cc62f47153453/html5/thumbnails/105.jpg)
Name _____________________________ Class ___________________Date ____________
Practice 8-6 Natural Logarithms Remember that common logarithms are logarithms of base 10. 4 4
10log3 log 3x x+ +=
e is the base of the Natural Logarithms, often abbreviated as ln. ( ) ( )log ln xe x =
Often called Euler’s number, e is an irrational that has a value of 2.718281828459045…
Changing loge x y= to exponential form would give ye x= .
Evaluating loge x There are two keys on the TI-84 that are for evaluating exponential functions with base e. They are the second function of the / (e) and L (ex) keys. Evaluate each expression to the nearest thousandth.
1. 5e ________________ 2. 4e− ________________ 3. 1
3e ________________ The same properties (product, quotient and power) of exponents apply to natural logarithms. So 33ln ln lnx y x y+ = Write as a single natural logarithm.
4. 4 ln 2 ln f− = ________________ 5. 1
lnx 3ln2
y+ = ________________
Solving natural logarithmic equations.
Solve ( )2ln 3 5 4x + =
Write in exponential form. ( )24 3 5e x= +
Take the square root of both sides. 4 3 5e x± = +
Subtract 5 from both sides. 4 5 3e x± − =
Divide both sides by 3. 4 5
3
ex
± − =
Evaluate using the calculator. 7.39 and 4.130x x= − =
6. Solve ln 9 5x = 7. Solve 2
ln 123
x + =
![Page 106: Practice 8-6 Natural Logarithmsstaff.bbhcsd.org/bradacm/files/2014/04/8.6-algebra-2.pdfApr 08, 2014 · Changing log e x y= to exponential form would give e xy =. Evaluating log e](https://reader035.fdocuments.in/reader035/viewer/2022070911/5fb18f72f20cc62f47153453/html5/thumbnails/106.jpg)
Solving an exponential equation. Solve 27 2.5 20xe + = Subtract 2.5 from both sides to start 27 17.5xe = isolating the exponential term. Divide both sides by 7 to finish 2 2.5xe = isolating the exponential term. Take the natural log of both sides. 2ln ln 2.5xe = Use the power property. 2 x ln ln 2.5e = Simplify using ln 1e = . 2 x ln 2.5=
Divide by 2. ln 2.5
x2
=
Use the calculator to finish. x 0.458≈ 8. Solve 1 30xe + = 9. Solve 34 1.2 14xe + =
Solve each equation. Round to the nearest ten-thousandth. Check your answer. Show all work for each. Circle your answer. 10. 18xe = 11. 2 12xe = 12. ln 3 6x =
13. ( )ln 4 1 36x − =
14. 5 4 7x
e + = 15. 22ln 2 x 1=
lne = 1
![Page 107: Practice 8-6 Natural Logarithmsstaff.bbhcsd.org/bradacm/files/2014/04/8.6-algebra-2.pdfApr 08, 2014 · Changing log e x y= to exponential form would give e xy =. Evaluating log e](https://reader035.fdocuments.in/reader035/viewer/2022070911/5fb18f72f20cc62f47153453/html5/thumbnails/107.jpg)
Name _____________________________ Class ___________________Date ____________
Practice 8-6 Natural Logarithms Remember that common logarithms are logarithms of base 10. 4 4
10log3 log 3x x+ +=
e is the base of the Natural Logarithms, often abbreviated as ln. ( ) ( )log ln xe x =
Often called Euler’s number, e is an irrational that has a value of 2.718281828459045…
Changing loge x y= to exponential form would give ye x= .
Evaluating loge x There are two keys on the TI-84 that are for evaluating exponential functions with base e. They are the second function of the / (e) and L (ex) keys. Evaluate each expression to the nearest thousandth.
1. 5e ________________ 2. 4e− ________________ 3. 1
3e ________________ The same properties (product, quotient and power) of exponents apply to natural logarithms. So 33ln ln lnx y x y+ = Write as a single natural logarithm.
4. 4 ln 2 ln f− = ________________ 5. 1
lnx 3ln2
y+ = ________________
Solving natural logarithmic equations.
Solve ( )2ln 3 5 4x + =
Write in exponential form. ( )24 3 5e x= +
Take the square root of both sides. 4 3 5e x± = +
Subtract 5 from both sides. 4 5 3e x± − =
Divide both sides by 3. 4 5
3
ex
± − =
Evaluate using the calculator. 7.39 and 4.130x x= − =
6. Solve ln 9 5x = 7. Solve 2
ln 123
x + =
![Page 108: Practice 8-6 Natural Logarithmsstaff.bbhcsd.org/bradacm/files/2014/04/8.6-algebra-2.pdfApr 08, 2014 · Changing log e x y= to exponential form would give e xy =. Evaluating log e](https://reader035.fdocuments.in/reader035/viewer/2022070911/5fb18f72f20cc62f47153453/html5/thumbnails/108.jpg)
Solving an exponential equation. Solve 27 2.5 20xe + = Subtract 2.5 from both sides to start 27 17.5xe = isolating the exponential term. Divide both sides by 7 to finish 2 2.5xe = isolating the exponential term. Take the natural log of both sides. 2ln ln 2.5xe = Use the power property. 2 x ln ln 2.5e = Simplify using ln 1e = . 2 x ln 2.5=
Divide by 2. ln 2.5
x2
=
Use the calculator to finish. x 0.458≈ 8. Solve 1 30xe + = 9. Solve 34 1.2 14xe + =
Solve each equation. Round to the nearest ten-thousandth. Check your answer. Show all work for each. Circle your answer. 10. 18xe = 11. 2 12xe = 12. ln 3 6x =
13. ( )ln 4 1 36x − =
14. 5 4 7x
e + = 15. 22ln 2 x 1=
lne = 1
![Page 109: Practice 8-6 Natural Logarithmsstaff.bbhcsd.org/bradacm/files/2014/04/8.6-algebra-2.pdfApr 08, 2014 · Changing log e x y= to exponential form would give e xy =. Evaluating log e](https://reader035.fdocuments.in/reader035/viewer/2022070911/5fb18f72f20cc62f47153453/html5/thumbnails/109.jpg)
Name _____________________________ Class ___________________Date ____________
Practice 8-6 Natural Logarithms Remember that common logarithms are logarithms of base 10. 4 4
10log3 log 3x x+ +=
e is the base of the Natural Logarithms, often abbreviated as ln. ( ) ( )log ln xe x =
Often called Euler’s number, e is an irrational that has a value of 2.718281828459045…
Changing loge x y= to exponential form would give ye x= .
Evaluating loge x There are two keys on the TI-84 that are for evaluating exponential functions with base e. They are the second function of the / (e) and L (ex) keys. Evaluate each expression to the nearest thousandth.
1. 5e ________________ 2. 4e− ________________ 3. 1
3e ________________ The same properties (product, quotient and power) of exponents apply to natural logarithms. So 33ln ln lnx y x y+ = Write as a single natural logarithm.
4. 4 ln 2 ln f− = ________________ 5. 1
lnx 3ln2
y+ = ________________
Solving natural logarithmic equations.
Solve ( )2ln 3 5 4x + =
Write in exponential form. ( )24 3 5e x= +
Take the square root of both sides. 4 3 5e x± = +
Subtract 5 from both sides. 4 5 3e x± − =
Divide both sides by 3. 4 5
3
ex
± − =
Evaluate using the calculator. 7.39 and 4.130x x= − =
6. Solve ln 9 5x = 7. Solve 2
ln 123
x + =
![Page 110: Practice 8-6 Natural Logarithmsstaff.bbhcsd.org/bradacm/files/2014/04/8.6-algebra-2.pdfApr 08, 2014 · Changing log e x y= to exponential form would give e xy =. Evaluating log e](https://reader035.fdocuments.in/reader035/viewer/2022070911/5fb18f72f20cc62f47153453/html5/thumbnails/110.jpg)
Solving an exponential equation. Solve 27 2.5 20xe + = Subtract 2.5 from both sides to start 27 17.5xe = isolating the exponential term. Divide both sides by 7 to finish 2 2.5xe = isolating the exponential term. Take the natural log of both sides. 2ln ln 2.5xe = Use the power property. 2 x ln ln 2.5e = Simplify using ln 1e = . 2 x ln 2.5=
Divide by 2. ln 2.5
x2
=
Use the calculator to finish. x 0.458≈ 8. Solve 1 30xe + = 9. Solve 34 1.2 14xe + =
Solve each equation. Round to the nearest ten-thousandth. Check your answer. Show all work for each. Circle your answer. 10. 18xe = 11. 2 12xe = 12. ln 3 6x =
13. ( )ln 4 1 36x − =
14. 5 4 7x
e + = 15. 22ln 2 x 1=
lne = 1
![Page 111: Practice 8-6 Natural Logarithmsstaff.bbhcsd.org/bradacm/files/2014/04/8.6-algebra-2.pdfApr 08, 2014 · Changing log e x y= to exponential form would give e xy =. Evaluating log e](https://reader035.fdocuments.in/reader035/viewer/2022070911/5fb18f72f20cc62f47153453/html5/thumbnails/111.jpg)
Name _____________________________ Class ___________________Date ____________
Practice 8-6 Natural Logarithms Remember that common logarithms are logarithms of base 10. 4 4
10log3 log 3x x+ +=
e is the base of the Natural Logarithms, often abbreviated as ln. ( ) ( )log ln xe x =
Often called Euler’s number, e is an irrational that has a value of 2.718281828459045…
Changing loge x y= to exponential form would give ye x= .
Evaluating loge x There are two keys on the TI-84 that are for evaluating exponential functions with base e. They are the second function of the / (e) and L (ex) keys. Evaluate each expression to the nearest thousandth.
1. 5e ________________ 2. 4e− ________________ 3. 1
3e ________________ The same properties (product, quotient and power) of exponents apply to natural logarithms. So 33ln ln lnx y x y+ = Write as a single natural logarithm.
4. 4 ln 2 ln f− = ________________ 5. 1
lnx 3ln2
y+ = ________________
Solving natural logarithmic equations.
Solve ( )2ln 3 5 4x + =
Write in exponential form. ( )24 3 5e x= +
Take the square root of both sides. 4 3 5e x± = +
Subtract 5 from both sides. 4 5 3e x± − =
Divide both sides by 3. 4 5
3
ex
± − =
Evaluate using the calculator. 7.39 and 4.130x x= − =
6. Solve ln 9 5x = 7. Solve 2
ln 123
x + =
![Page 112: Practice 8-6 Natural Logarithmsstaff.bbhcsd.org/bradacm/files/2014/04/8.6-algebra-2.pdfApr 08, 2014 · Changing log e x y= to exponential form would give e xy =. Evaluating log e](https://reader035.fdocuments.in/reader035/viewer/2022070911/5fb18f72f20cc62f47153453/html5/thumbnails/112.jpg)
Solving an exponential equation. Solve 27 2.5 20xe + = Subtract 2.5 from both sides to start 27 17.5xe = isolating the exponential term. Divide both sides by 7 to finish 2 2.5xe = isolating the exponential term. Take the natural log of both sides. 2ln ln 2.5xe = Use the power property. 2 x ln ln 2.5e = Simplify using ln 1e = . 2 x ln 2.5=
Divide by 2. ln 2.5
x2
=
Use the calculator to finish. x 0.458≈ 8. Solve 1 30xe + = 9. Solve 34 1.2 14xe + =
Solve each equation. Round to the nearest ten-thousandth. Check your answer. Show all work for each. Circle your answer. 10. 18xe = 11. 2 12xe = 12. ln 3 6x =
13. ( )ln 4 1 36x − =
14. 5 4 7x
e + = 15. 22ln 2 x 1=
lne = 1
![Page 113: Practice 8-6 Natural Logarithmsstaff.bbhcsd.org/bradacm/files/2014/04/8.6-algebra-2.pdfApr 08, 2014 · Changing log e x y= to exponential form would give e xy =. Evaluating log e](https://reader035.fdocuments.in/reader035/viewer/2022070911/5fb18f72f20cc62f47153453/html5/thumbnails/113.jpg)
Name _____________________________ Class ___________________Date ____________
Practice 8-6 Natural Logarithms Remember that common logarithms are logarithms of base 10. 4 4
10log3 log 3x x+ +=
e is the base of the Natural Logarithms, often abbreviated as ln. ( ) ( )log ln xe x =
Often called Euler’s number, e is an irrational that has a value of 2.718281828459045…
Changing loge x y= to exponential form would give ye x= .
Evaluating loge x There are two keys on the TI-84 that are for evaluating exponential functions with base e. They are the second function of the / (e) and L (ex) keys. Evaluate each expression to the nearest thousandth.
1. 5e ________________ 2. 4e− ________________ 3. 1
3e ________________ The same properties (product, quotient and power) of exponents apply to natural logarithms. So 33ln ln lnx y x y+ = Write as a single natural logarithm.
4. 4 ln 2 ln f− = ________________ 5. 1
lnx 3ln2
y+ = ________________
Solving natural logarithmic equations.
Solve ( )2ln 3 5 4x + =
Write in exponential form. ( )24 3 5e x= +
Take the square root of both sides. 4 3 5e x± = +
Subtract 5 from both sides. 4 5 3e x± − =
Divide both sides by 3. 4 5
3
ex
± − =
Evaluate using the calculator. 7.39 and 4.130x x= − =
6. Solve ln 9 5x = 7. Solve 2
ln 123
x + =
![Page 114: Practice 8-6 Natural Logarithmsstaff.bbhcsd.org/bradacm/files/2014/04/8.6-algebra-2.pdfApr 08, 2014 · Changing log e x y= to exponential form would give e xy =. Evaluating log e](https://reader035.fdocuments.in/reader035/viewer/2022070911/5fb18f72f20cc62f47153453/html5/thumbnails/114.jpg)
Solving an exponential equation. Solve 27 2.5 20xe + = Subtract 2.5 from both sides to start 27 17.5xe = isolating the exponential term. Divide both sides by 7 to finish 2 2.5xe = isolating the exponential term. Take the natural log of both sides. 2ln ln 2.5xe = Use the power property. 2 x ln ln 2.5e = Simplify using ln 1e = . 2 x ln 2.5=
Divide by 2. ln 2.5
x2
=
Use the calculator to finish. x 0.458≈ 8. Solve 1 30xe + = 9. Solve 34 1.2 14xe + =
Solve each equation. Round to the nearest ten-thousandth. Check your answer. Show all work for each. Circle your answer. 10. 18xe = 11. 2 12xe = 12. ln 3 6x =
13. ( )ln 4 1 36x − =
14. 5 4 7x
e + = 15. 22ln 2 x 1=
lne = 1
![Page 115: Practice 8-6 Natural Logarithmsstaff.bbhcsd.org/bradacm/files/2014/04/8.6-algebra-2.pdfApr 08, 2014 · Changing log e x y= to exponential form would give e xy =. Evaluating log e](https://reader035.fdocuments.in/reader035/viewer/2022070911/5fb18f72f20cc62f47153453/html5/thumbnails/115.jpg)
Name _____________________________ Class ___________________Date ____________
Practice 8-6 Natural Logarithms Remember that common logarithms are logarithms of base 10. 4 4
10log3 log 3x x+ +=
e is the base of the Natural Logarithms, often abbreviated as ln. ( ) ( )log ln xe x =
Often called Euler’s number, e is an irrational that has a value of 2.718281828459045…
Changing loge x y= to exponential form would give ye x= .
Evaluating loge x There are two keys on the TI-84 that are for evaluating exponential functions with base e. They are the second function of the / (e) and L (ex) keys. Evaluate each expression to the nearest thousandth.
1. 5e ________________ 2. 4e− ________________ 3. 1
3e ________________ The same properties (product, quotient and power) of exponents apply to natural logarithms. So 33ln ln lnx y x y+ = Write as a single natural logarithm.
4. 4 ln 2 ln f− = ________________ 5. 1
lnx 3ln2
y+ = ________________
Solving natural logarithmic equations.
Solve ( )2ln 3 5 4x + =
Write in exponential form. ( )24 3 5e x= +
Take the square root of both sides. 4 3 5e x± = +
Subtract 5 from both sides. 4 5 3e x± − =
Divide both sides by 3. 4 5
3
ex
± − =
Evaluate using the calculator. 7.39 and 4.130x x= − =
6. Solve ln 9 5x = 7. Solve 2
ln 123
x + =
![Page 116: Practice 8-6 Natural Logarithmsstaff.bbhcsd.org/bradacm/files/2014/04/8.6-algebra-2.pdfApr 08, 2014 · Changing log e x y= to exponential form would give e xy =. Evaluating log e](https://reader035.fdocuments.in/reader035/viewer/2022070911/5fb18f72f20cc62f47153453/html5/thumbnails/116.jpg)
Solving an exponential equation. Solve 27 2.5 20xe + = Subtract 2.5 from both sides to start 27 17.5xe = isolating the exponential term. Divide both sides by 7 to finish 2 2.5xe = isolating the exponential term. Take the natural log of both sides. 2ln ln 2.5xe = Use the power property. 2 x ln ln 2.5e = Simplify using ln 1e = . 2 x ln 2.5=
Divide by 2. ln 2.5
x2
=
Use the calculator to finish. x 0.458≈ 8. Solve 1 30xe + = 9. Solve 34 1.2 14xe + =
Solve each equation. Round to the nearest ten-thousandth. Check your answer. Show all work for each. Circle your answer. 10. 18xe = 11. 2 12xe = 12. ln 3 6x =
13. ( )ln 4 1 36x − =
14. 5 4 7x
e + = 15. 22ln 2 x 1=
lne = 1
![Page 117: Practice 8-6 Natural Logarithmsstaff.bbhcsd.org/bradacm/files/2014/04/8.6-algebra-2.pdfApr 08, 2014 · Changing log e x y= to exponential form would give e xy =. Evaluating log e](https://reader035.fdocuments.in/reader035/viewer/2022070911/5fb18f72f20cc62f47153453/html5/thumbnails/117.jpg)
Name _____________________________ Class ___________________Date ____________
Practice 8-6 Natural Logarithms Remember that common logarithms are logarithms of base 10. 4 4
10log3 log 3x x+ +=
e is the base of the Natural Logarithms, often abbreviated as ln. ( ) ( )log ln xe x =
Often called Euler’s number, e is an irrational that has a value of 2.718281828459045…
Changing loge x y= to exponential form would give ye x= .
Evaluating loge x There are two keys on the TI-84 that are for evaluating exponential functions with base e. They are the second function of the / (e) and L (ex) keys. Evaluate each expression to the nearest thousandth.
1. 5e ________________ 2. 4e− ________________ 3. 1
3e ________________ The same properties (product, quotient and power) of exponents apply to natural logarithms. So 33ln ln lnx y x y+ = Write as a single natural logarithm.
4. 4 ln 2 ln f− = ________________ 5. 1
lnx 3ln2
y+ = ________________
Solving natural logarithmic equations.
Solve ( )2ln 3 5 4x + =
Write in exponential form. ( )24 3 5e x= +
Take the square root of both sides. 4 3 5e x± = +
Subtract 5 from both sides. 4 5 3e x± − =
Divide both sides by 3. 4 5
3
ex
± − =
Evaluate using the calculator. 7.39 and 4.130x x= − =
6. Solve ln 9 5x = 7. Solve 2
ln 123
x + =
![Page 118: Practice 8-6 Natural Logarithmsstaff.bbhcsd.org/bradacm/files/2014/04/8.6-algebra-2.pdfApr 08, 2014 · Changing log e x y= to exponential form would give e xy =. Evaluating log e](https://reader035.fdocuments.in/reader035/viewer/2022070911/5fb18f72f20cc62f47153453/html5/thumbnails/118.jpg)
Solving an exponential equation. Solve 27 2.5 20xe + = Subtract 2.5 from both sides to start 27 17.5xe = isolating the exponential term. Divide both sides by 7 to finish 2 2.5xe = isolating the exponential term. Take the natural log of both sides. 2ln ln 2.5xe = Use the power property. 2 x ln ln 2.5e = Simplify using ln 1e = . 2 x ln 2.5=
Divide by 2. ln 2.5
x2
=
Use the calculator to finish. x 0.458≈ 8. Solve 1 30xe + = 9. Solve 34 1.2 14xe + =
Solve each equation. Round to the nearest ten-thousandth. Check your answer. Show all work for each. Circle your answer. 10. 18xe = 11. 2 12xe = 12. ln 3 6x =
13. ( )ln 4 1 36x − =
14. 5 4 7x
e + = 15. 22ln 2 x 1=
lne = 1
![Page 119: Practice 8-6 Natural Logarithmsstaff.bbhcsd.org/bradacm/files/2014/04/8.6-algebra-2.pdfApr 08, 2014 · Changing log e x y= to exponential form would give e xy =. Evaluating log e](https://reader035.fdocuments.in/reader035/viewer/2022070911/5fb18f72f20cc62f47153453/html5/thumbnails/119.jpg)
Name _____________________________ Class ___________________Date ____________
Practice 8-6 Natural Logarithms Remember that common logarithms are logarithms of base 10. 4 4
10log3 log 3x x+ +=
e is the base of the Natural Logarithms, often abbreviated as ln. ( ) ( )log ln xe x =
Often called Euler’s number, e is an irrational that has a value of 2.718281828459045…
Changing loge x y= to exponential form would give ye x= .
Evaluating loge x There are two keys on the TI-84 that are for evaluating exponential functions with base e. They are the second function of the / (e) and L (ex) keys. Evaluate each expression to the nearest thousandth.
1. 5e ________________ 2. 4e− ________________ 3. 1
3e ________________ The same properties (product, quotient and power) of exponents apply to natural logarithms. So 33ln ln lnx y x y+ = Write as a single natural logarithm.
4. 4 ln 2 ln f− = ________________ 5. 1
lnx 3ln2
y+ = ________________
Solving natural logarithmic equations.
Solve ( )2ln 3 5 4x + =
Write in exponential form. ( )24 3 5e x= +
Take the square root of both sides. 4 3 5e x± = +
Subtract 5 from both sides. 4 5 3e x± − =
Divide both sides by 3. 4 5
3
ex
± − =
Evaluate using the calculator. 7.39 and 4.130x x= − =
6. Solve ln 9 5x = 7. Solve 2
ln 123
x + =
![Page 120: Practice 8-6 Natural Logarithmsstaff.bbhcsd.org/bradacm/files/2014/04/8.6-algebra-2.pdfApr 08, 2014 · Changing log e x y= to exponential form would give e xy =. Evaluating log e](https://reader035.fdocuments.in/reader035/viewer/2022070911/5fb18f72f20cc62f47153453/html5/thumbnails/120.jpg)
Solving an exponential equation. Solve 27 2.5 20xe + = Subtract 2.5 from both sides to start 27 17.5xe = isolating the exponential term. Divide both sides by 7 to finish 2 2.5xe = isolating the exponential term. Take the natural log of both sides. 2ln ln 2.5xe = Use the power property. 2 x ln ln 2.5e = Simplify using ln 1e = . 2 x ln 2.5=
Divide by 2. ln 2.5
x2
=
Use the calculator to finish. x 0.458≈ 8. Solve 1 30xe + = 9. Solve 34 1.2 14xe + =
Solve each equation. Round to the nearest ten-thousandth. Check your answer. Show all work for each. Circle your answer. 10. 18xe = 11. 2 12xe = 12. ln 3 6x =
13. ( )ln 4 1 36x − =
14. 5 4 7x
e + = 15. 22ln 2 x 1=
lne = 1
![Page 121: Practice 8-6 Natural Logarithmsstaff.bbhcsd.org/bradacm/files/2014/04/8.6-algebra-2.pdfApr 08, 2014 · Changing log e x y= to exponential form would give e xy =. Evaluating log e](https://reader035.fdocuments.in/reader035/viewer/2022070911/5fb18f72f20cc62f47153453/html5/thumbnails/121.jpg)
Name _____________________________ Class ___________________Date ____________
Practice 8-6 Natural Logarithms Remember that common logarithms are logarithms of base 10. 4 4
10log3 log 3x x+ +=
e is the base of the Natural Logarithms, often abbreviated as ln. ( ) ( )log ln xe x =
Often called Euler’s number, e is an irrational that has a value of 2.718281828459045…
Changing loge x y= to exponential form would give ye x= .
Evaluating loge x There are two keys on the TI-84 that are for evaluating exponential functions with base e. They are the second function of the / (e) and L (ex) keys. Evaluate each expression to the nearest thousandth.
1. 5e ________________ 2. 4e− ________________ 3. 1
3e ________________ The same properties (product, quotient and power) of exponents apply to natural logarithms. So 33ln ln lnx y x y+ = Write as a single natural logarithm.
4. 4 ln 2 ln f− = ________________ 5. 1
lnx 3ln2
y+ = ________________
Solving natural logarithmic equations.
Solve ( )2ln 3 5 4x + =
Write in exponential form. ( )24 3 5e x= +
Take the square root of both sides. 4 3 5e x± = +
Subtract 5 from both sides. 4 5 3e x± − =
Divide both sides by 3. 4 5
3
ex
± − =
Evaluate using the calculator. 7.39 and 4.130x x= − =
6. Solve ln 9 5x = 7. Solve 2
ln 123
x + =
![Page 122: Practice 8-6 Natural Logarithmsstaff.bbhcsd.org/bradacm/files/2014/04/8.6-algebra-2.pdfApr 08, 2014 · Changing log e x y= to exponential form would give e xy =. Evaluating log e](https://reader035.fdocuments.in/reader035/viewer/2022070911/5fb18f72f20cc62f47153453/html5/thumbnails/122.jpg)
Solving an exponential equation. Solve 27 2.5 20xe + = Subtract 2.5 from both sides to start 27 17.5xe = isolating the exponential term. Divide both sides by 7 to finish 2 2.5xe = isolating the exponential term. Take the natural log of both sides. 2ln ln 2.5xe = Use the power property. 2 x ln ln 2.5e = Simplify using ln 1e = . 2 x ln 2.5=
Divide by 2. ln 2.5
x2
=
Use the calculator to finish. x 0.458≈ 8. Solve 1 30xe + = 9. Solve 34 1.2 14xe + =
Solve each equation. Round to the nearest ten-thousandth. Check your answer. Show all work for each. Circle your answer. 10. 18xe = 11. 2 12xe = 12. ln 3 6x =
13. ( )ln 4 1 36x − =
14. 5 4 7x
e + = 15. 22ln 2 x 1=
lne = 1
![Page 123: Practice 8-6 Natural Logarithmsstaff.bbhcsd.org/bradacm/files/2014/04/8.6-algebra-2.pdfApr 08, 2014 · Changing log e x y= to exponential form would give e xy =. Evaluating log e](https://reader035.fdocuments.in/reader035/viewer/2022070911/5fb18f72f20cc62f47153453/html5/thumbnails/123.jpg)
Name _____________________________ Class ___________________Date ____________
Practice 8-6 Natural Logarithms Remember that common logarithms are logarithms of base 10. 4 4
10log3 log 3x x+ +=
e is the base of the Natural Logarithms, often abbreviated as ln. ( ) ( )log ln xe x =
Often called Euler’s number, e is an irrational that has a value of 2.718281828459045…
Changing loge x y= to exponential form would give ye x= .
Evaluating loge x There are two keys on the TI-84 that are for evaluating exponential functions with base e. They are the second function of the / (e) and L (ex) keys. Evaluate each expression to the nearest thousandth.
1. 5e ________________ 2. 4e− ________________ 3. 1
3e ________________ The same properties (product, quotient and power) of exponents apply to natural logarithms. So 33ln ln lnx y x y+ = Write as a single natural logarithm.
4. 4 ln 2 ln f− = ________________ 5. 1
lnx 3ln2
y+ = ________________
Solving natural logarithmic equations.
Solve ( )2ln 3 5 4x + =
Write in exponential form. ( )24 3 5e x= +
Take the square root of both sides. 4 3 5e x± = +
Subtract 5 from both sides. 4 5 3e x± − =
Divide both sides by 3. 4 5
3
ex
± − =
Evaluate using the calculator. 7.39 and 4.130x x= − =
6. Solve ln 9 5x = 7. Solve 2
ln 123
x + =
![Page 124: Practice 8-6 Natural Logarithmsstaff.bbhcsd.org/bradacm/files/2014/04/8.6-algebra-2.pdfApr 08, 2014 · Changing log e x y= to exponential form would give e xy =. Evaluating log e](https://reader035.fdocuments.in/reader035/viewer/2022070911/5fb18f72f20cc62f47153453/html5/thumbnails/124.jpg)
Solving an exponential equation. Solve 27 2.5 20xe + = Subtract 2.5 from both sides to start 27 17.5xe = isolating the exponential term. Divide both sides by 7 to finish 2 2.5xe = isolating the exponential term. Take the natural log of both sides. 2ln ln 2.5xe = Use the power property. 2 x ln ln 2.5e = Simplify using ln 1e = . 2 x ln 2.5=
Divide by 2. ln 2.5
x2
=
Use the calculator to finish. x 0.458≈ 8. Solve 1 30xe + = 9. Solve 34 1.2 14xe + =
Solve each equation. Round to the nearest ten-thousandth. Check your answer. Show all work for each. Circle your answer. 10. 18xe = 11. 2 12xe = 12. ln 3 6x =
13. ( )ln 4 1 36x − =
14. 5 4 7x
e + = 15. 22ln 2 x 1=
lne = 1
![Page 125: Practice 8-6 Natural Logarithmsstaff.bbhcsd.org/bradacm/files/2014/04/8.6-algebra-2.pdfApr 08, 2014 · Changing log e x y= to exponential form would give e xy =. Evaluating log e](https://reader035.fdocuments.in/reader035/viewer/2022070911/5fb18f72f20cc62f47153453/html5/thumbnails/125.jpg)
Name _____________________________ Class ___________________Date ____________
Practice 8-6 Natural Logarithms Remember that common logarithms are logarithms of base 10. 4 4
10log3 log 3x x+ +=
e is the base of the Natural Logarithms, often abbreviated as ln. ( ) ( )log ln xe x =
Often called Euler’s number, e is an irrational that has a value of 2.718281828459045…
Changing loge x y= to exponential form would give ye x= .
Evaluating loge x There are two keys on the TI-84 that are for evaluating exponential functions with base e. They are the second function of the / (e) and L (ex) keys. Evaluate each expression to the nearest thousandth.
1. 5e ________________ 2. 4e− ________________ 3. 1
3e ________________ The same properties (product, quotient and power) of exponents apply to natural logarithms. So 33ln ln lnx y x y+ = Write as a single natural logarithm.
4. 4 ln 2 ln f− = ________________ 5. 1
lnx 3ln2
y+ = ________________
Solving natural logarithmic equations.
Solve ( )2ln 3 5 4x + =
Write in exponential form. ( )24 3 5e x= +
Take the square root of both sides. 4 3 5e x± = +
Subtract 5 from both sides. 4 5 3e x± − =
Divide both sides by 3. 4 5
3
ex
± − =
Evaluate using the calculator. 7.39 and 4.130x x= − =
6. Solve ln 9 5x = 7. Solve 2
ln 123
x + =
![Page 126: Practice 8-6 Natural Logarithmsstaff.bbhcsd.org/bradacm/files/2014/04/8.6-algebra-2.pdfApr 08, 2014 · Changing log e x y= to exponential form would give e xy =. Evaluating log e](https://reader035.fdocuments.in/reader035/viewer/2022070911/5fb18f72f20cc62f47153453/html5/thumbnails/126.jpg)
Solving an exponential equation. Solve 27 2.5 20xe + = Subtract 2.5 from both sides to start 27 17.5xe = isolating the exponential term. Divide both sides by 7 to finish 2 2.5xe = isolating the exponential term. Take the natural log of both sides. 2ln ln 2.5xe = Use the power property. 2 x ln ln 2.5e = Simplify using ln 1e = . 2 x ln 2.5=
Divide by 2. ln 2.5
x2
=
Use the calculator to finish. x 0.458≈ 8. Solve 1 30xe + = 9. Solve 34 1.2 14xe + =
Solve each equation. Round to the nearest ten-thousandth. Check your answer. Show all work for each. Circle your answer. 10. 18xe = 11. 2 12xe = 12. ln 3 6x =
13. ( )ln 4 1 36x − =
14. 5 4 7x
e + = 15. 22ln 2 x 1=
lne = 1
![Page 127: Practice 8-6 Natural Logarithmsstaff.bbhcsd.org/bradacm/files/2014/04/8.6-algebra-2.pdfApr 08, 2014 · Changing log e x y= to exponential form would give e xy =. Evaluating log e](https://reader035.fdocuments.in/reader035/viewer/2022070911/5fb18f72f20cc62f47153453/html5/thumbnails/127.jpg)
Name _____________________________ Class ___________________Date ____________
Practice 8-6 Natural Logarithms Remember that common logarithms are logarithms of base 10. 4 4
10log3 log 3x x+ +=
e is the base of the Natural Logarithms, often abbreviated as ln. ( ) ( )log ln xe x =
Often called Euler’s number, e is an irrational that has a value of 2.718281828459045…
Changing loge x y= to exponential form would give ye x= .
Evaluating loge x There are two keys on the TI-84 that are for evaluating exponential functions with base e. They are the second function of the / (e) and L (ex) keys. Evaluate each expression to the nearest thousandth.
1. 5e ________________ 2. 4e− ________________ 3. 1
3e ________________ The same properties (product, quotient and power) of exponents apply to natural logarithms. So 33ln ln lnx y x y+ = Write as a single natural logarithm.
4. 4 ln 2 ln f− = ________________ 5. 1
lnx 3ln2
y+ = ________________
Solving natural logarithmic equations.
Solve ( )2ln 3 5 4x + =
Write in exponential form. ( )24 3 5e x= +
Take the square root of both sides. 4 3 5e x± = +
Subtract 5 from both sides. 4 5 3e x± − =
Divide both sides by 3. 4 5
3
ex
± − =
Evaluate using the calculator. 7.39 and 4.130x x= − =
6. Solve ln 9 5x = 7. Solve 2
ln 123
x + =
![Page 128: Practice 8-6 Natural Logarithmsstaff.bbhcsd.org/bradacm/files/2014/04/8.6-algebra-2.pdfApr 08, 2014 · Changing log e x y= to exponential form would give e xy =. Evaluating log e](https://reader035.fdocuments.in/reader035/viewer/2022070911/5fb18f72f20cc62f47153453/html5/thumbnails/128.jpg)
Solving an exponential equation. Solve 27 2.5 20xe + = Subtract 2.5 from both sides to start 27 17.5xe = isolating the exponential term. Divide both sides by 7 to finish 2 2.5xe = isolating the exponential term. Take the natural log of both sides. 2ln ln 2.5xe = Use the power property. 2 x ln ln 2.5e = Simplify using ln 1e = . 2 x ln 2.5=
Divide by 2. ln 2.5
x2
=
Use the calculator to finish. x 0.458≈ 8. Solve 1 30xe + = 9. Solve 34 1.2 14xe + =
Solve each equation. Round to the nearest ten-thousandth. Check your answer. Show all work for each. Circle your answer. 10. 18xe = 11. 2 12xe = 12. ln 3 6x =
13. ( )ln 4 1 36x − =
14. 5 4 7x
e + = 15. 22ln 2 x 1=
lne = 1
![Page 129: Practice 8-6 Natural Logarithmsstaff.bbhcsd.org/bradacm/files/2014/04/8.6-algebra-2.pdfApr 08, 2014 · Changing log e x y= to exponential form would give e xy =. Evaluating log e](https://reader035.fdocuments.in/reader035/viewer/2022070911/5fb18f72f20cc62f47153453/html5/thumbnails/129.jpg)
Name _____________________________ Class ___________________Date ____________
Practice 8-6 Natural Logarithms Remember that common logarithms are logarithms of base 10. 4 4
10log3 log 3x x+ +=
e is the base of the Natural Logarithms, often abbreviated as ln. ( ) ( )log ln xe x =
Often called Euler’s number, e is an irrational that has a value of 2.718281828459045…
Changing loge x y= to exponential form would give ye x= .
Evaluating loge x There are two keys on the TI-84 that are for evaluating exponential functions with base e. They are the second function of the / (e) and L (ex) keys. Evaluate each expression to the nearest thousandth.
1. 5e ________________ 2. 4e− ________________ 3. 1
3e ________________ The same properties (product, quotient and power) of exponents apply to natural logarithms. So 33ln ln lnx y x y+ = Write as a single natural logarithm.
4. 4 ln 2 ln f− = ________________ 5. 1
lnx 3ln2
y+ = ________________
Solving natural logarithmic equations.
Solve ( )2ln 3 5 4x + =
Write in exponential form. ( )24 3 5e x= +
Take the square root of both sides. 4 3 5e x± = +
Subtract 5 from both sides. 4 5 3e x± − =
Divide both sides by 3. 4 5
3
ex
± − =
Evaluate using the calculator. 7.39 and 4.130x x= − =
6. Solve ln 9 5x = 7. Solve 2
ln 123
x + =
![Page 130: Practice 8-6 Natural Logarithmsstaff.bbhcsd.org/bradacm/files/2014/04/8.6-algebra-2.pdfApr 08, 2014 · Changing log e x y= to exponential form would give e xy =. Evaluating log e](https://reader035.fdocuments.in/reader035/viewer/2022070911/5fb18f72f20cc62f47153453/html5/thumbnails/130.jpg)
Solving an exponential equation. Solve 27 2.5 20xe + = Subtract 2.5 from both sides to start 27 17.5xe = isolating the exponential term. Divide both sides by 7 to finish 2 2.5xe = isolating the exponential term. Take the natural log of both sides. 2ln ln 2.5xe = Use the power property. 2 x ln ln 2.5e = Simplify using ln 1e = . 2 x ln 2.5=
Divide by 2. ln 2.5
x2
=
Use the calculator to finish. x 0.458≈ 8. Solve 1 30xe + = 9. Solve 34 1.2 14xe + =
Solve each equation. Round to the nearest ten-thousandth. Check your answer. Show all work for each. Circle your answer. 10. 18xe = 11. 2 12xe = 12. ln 3 6x =
13. ( )ln 4 1 36x − =
14. 5 4 7x
e + = 15. 22ln 2 x 1=
lne = 1
![Page 131: Practice 8-6 Natural Logarithmsstaff.bbhcsd.org/bradacm/files/2014/04/8.6-algebra-2.pdfApr 08, 2014 · Changing log e x y= to exponential form would give e xy =. Evaluating log e](https://reader035.fdocuments.in/reader035/viewer/2022070911/5fb18f72f20cc62f47153453/html5/thumbnails/131.jpg)
Name _____________________________ Class ___________________Date ____________
Practice 8-6 Natural Logarithms Remember that common logarithms are logarithms of base 10. 4 4
10log3 log 3x x+ +=
e is the base of the Natural Logarithms, often abbreviated as ln. ( ) ( )log ln xe x =
Often called Euler’s number, e is an irrational that has a value of 2.718281828459045…
Changing loge x y= to exponential form would give ye x= .
Evaluating loge x There are two keys on the TI-84 that are for evaluating exponential functions with base e. They are the second function of the / (e) and L (ex) keys. Evaluate each expression to the nearest thousandth.
1. 5e ________________ 2. 4e− ________________ 3. 1
3e ________________ The same properties (product, quotient and power) of exponents apply to natural logarithms. So 33ln ln lnx y x y+ = Write as a single natural logarithm.
4. 4 ln 2 ln f− = ________________ 5. 1
lnx 3ln2
y+ = ________________
Solving natural logarithmic equations.
Solve ( )2ln 3 5 4x + =
Write in exponential form. ( )24 3 5e x= +
Take the square root of both sides. 4 3 5e x± = +
Subtract 5 from both sides. 4 5 3e x± − =
Divide both sides by 3. 4 5
3
ex
± − =
Evaluate using the calculator. 7.39 and 4.130x x= − =
6. Solve ln 9 5x = 7. Solve 2
ln 123
x + =
![Page 132: Practice 8-6 Natural Logarithmsstaff.bbhcsd.org/bradacm/files/2014/04/8.6-algebra-2.pdfApr 08, 2014 · Changing log e x y= to exponential form would give e xy =. Evaluating log e](https://reader035.fdocuments.in/reader035/viewer/2022070911/5fb18f72f20cc62f47153453/html5/thumbnails/132.jpg)
Solving an exponential equation. Solve 27 2.5 20xe + = Subtract 2.5 from both sides to start 27 17.5xe = isolating the exponential term. Divide both sides by 7 to finish 2 2.5xe = isolating the exponential term. Take the natural log of both sides. 2ln ln 2.5xe = Use the power property. 2 x ln ln 2.5e = Simplify using ln 1e = . 2 x ln 2.5=
Divide by 2. ln 2.5
x2
=
Use the calculator to finish. x 0.458≈ 8. Solve 1 30xe + = 9. Solve 34 1.2 14xe + =
Solve each equation. Round to the nearest ten-thousandth. Check your answer. Show all work for each. Circle your answer. 10. 18xe = 11. 2 12xe = 12. ln 3 6x =
13. ( )ln 4 1 36x − =
14. 5 4 7x
e + = 15. 22ln 2 x 1=
lne = 1
![Page 133: Practice 8-6 Natural Logarithmsstaff.bbhcsd.org/bradacm/files/2014/04/8.6-algebra-2.pdfApr 08, 2014 · Changing log e x y= to exponential form would give e xy =. Evaluating log e](https://reader035.fdocuments.in/reader035/viewer/2022070911/5fb18f72f20cc62f47153453/html5/thumbnails/133.jpg)
Name _____________________________ Class ___________________Date ____________
Practice 8-6 Natural Logarithms Remember that common logarithms are logarithms of base 10. 4 4
10log3 log 3x x+ +=
e is the base of the Natural Logarithms, often abbreviated as ln. ( ) ( )log ln xe x =
Often called Euler’s number, e is an irrational that has a value of 2.718281828459045…
Changing loge x y= to exponential form would give ye x= .
Evaluating loge x There are two keys on the TI-84 that are for evaluating exponential functions with base e. They are the second function of the / (e) and L (ex) keys. Evaluate each expression to the nearest thousandth.
1. 5e ________________ 2. 4e− ________________ 3. 1
3e ________________ The same properties (product, quotient and power) of exponents apply to natural logarithms. So 33ln ln lnx y x y+ = Write as a single natural logarithm.
4. 4 ln 2 ln f− = ________________ 5. 1
lnx 3ln2
y+ = ________________
Solving natural logarithmic equations.
Solve ( )2ln 3 5 4x + =
Write in exponential form. ( )24 3 5e x= +
Take the square root of both sides. 4 3 5e x± = +
Subtract 5 from both sides. 4 5 3e x± − =
Divide both sides by 3. 4 5
3
ex
± − =
Evaluate using the calculator. 7.39 and 4.130x x= − =
6. Solve ln 9 5x = 7. Solve 2
ln 123
x + =
![Page 134: Practice 8-6 Natural Logarithmsstaff.bbhcsd.org/bradacm/files/2014/04/8.6-algebra-2.pdfApr 08, 2014 · Changing log e x y= to exponential form would give e xy =. Evaluating log e](https://reader035.fdocuments.in/reader035/viewer/2022070911/5fb18f72f20cc62f47153453/html5/thumbnails/134.jpg)
Solving an exponential equation. Solve 27 2.5 20xe + = Subtract 2.5 from both sides to start 27 17.5xe = isolating the exponential term. Divide both sides by 7 to finish 2 2.5xe = isolating the exponential term. Take the natural log of both sides. 2ln ln 2.5xe = Use the power property. 2 x ln ln 2.5e = Simplify using ln 1e = . 2 x ln 2.5=
Divide by 2. ln 2.5
x2
=
Use the calculator to finish. x 0.458≈ 8. Solve 1 30xe + = 9. Solve 34 1.2 14xe + =
Solve each equation. Round to the nearest ten-thousandth. Check your answer. Show all work for each. Circle your answer. 10. 18xe = 11. 2 12xe = 12. ln 3 6x =
13. ( )ln 4 1 36x − =
14. 5 4 7x
e + = 15. 22ln 2 x 1=
lne = 1