CHANGE OF BASE FORMULA
6826.15log
15log15log5
b
aab log
loglog
Ex: Rewrite log515 using the change of base formula
Steps for solving exponential equations
Take a common logarithm of each side
Use the power property of logarithms
Solve for x by dividing Use a calculator to find the
approximate value
Solving Exponential Equations
43 x
43 x
4log3log x
3log
4logx
4log3log x 1. Take the log of both sides
2. Use the power property
3. Solve for x.
Solve . Round to the nearest ten-thousandth.
X=1.2619 4. Use a calculator.
Check your answer – 31.2619=4
Another Example
1013 4 x
1013 4 x
101log3log)4( x
3log
101log4 x
101log3log 4 x 1. Take the log of both sides
2. Use the power property
3. Solve for x.
Solve . Round to the nearest ten-thousandth.
X=4.2009 – 4 = 0.2009 4. Use a calculator.
Check your answer – 30.2009+4=101
CHANGE OF BASE – HOW IT WORKS
Use the change of base formula to evaluate . Then convert it to a logarithm of base 2.
15log3
4650.215log3
x23 log15log
3log
15log15log3 1. Rewrite using the
change of base formula
2. Use a calculator
3. Write an equation to convert to base 2
CHANGE OF BASE – HOW IT WORKS
x2log4650.2
2log
log4650.2
x
xlog7420.0
7420.010x
xlog2log 4650.2 6. Multiply both sides of the equation by log2
7. Use a calculator; simplify.
8. Write in exponential form.
5. Rewrite using the change of base formula
4. Substitute log315=2.4650
X=5.5208 9. Use a calculator.
Log315 is approximately equal to 2.4650 or log25.5208
Let’s try one
Use the change of base formula to evaluate . Then convert it to a logarithm of base 8.
400log5
7227.3400log5
x85 log400log
5log
400log400log5 1. Rewrite using the
change of base formula
2. Use a calculator
3. Write an equation to convert to base 2
x8log7227.3
8log
log7227.3
x
xlog3619.3
3619.310x
xlog8log .72273 6. Multiply both sides of the equation by log8
7. Use a calculator; simplify.
8. Write in exponential form.
5. Rewrite using the change of base formula
4. Substitute log5400=3.727
X=2301 9. Use a calculator.
Log5400 is approximately equal to 3.7227 or log82301
SOLVING SIMPLE LOG EQUATIONS
x642
3
8
6
16x
26log
232log
4
4
x
x
2)3(log2log solve tologs of properties Use 44 x
1. Use the product property
2. Write in exponential form.
x616
2)3(logx2log 44
3. Simplify
4. Solve for x.
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