5.3 Intro to Logarithms 2/27/2013. Definition of a Logarithmic Function For y > 0 and b > 0, b ≠...
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Transcript of 5.3 Intro to Logarithms 2/27/2013. Definition of a Logarithmic Function For y > 0 and b > 0, b ≠...
![Page 1: 5.3 Intro to Logarithms 2/27/2013. Definition of a Logarithmic Function For y > 0 and b > 0, b ≠ 1, log b y = x if and only if b x = y Note: Logarithmic.](https://reader035.fdocuments.in/reader035/viewer/2022062217/5697bfa81a28abf838c994aa/html5/thumbnails/1.jpg)
5.3 Intro to Logarithms
2/27/2013
![Page 2: 5.3 Intro to Logarithms 2/27/2013. Definition of a Logarithmic Function For y > 0 and b > 0, b ≠ 1, log b y = x if and only if b x = y Note: Logarithmic.](https://reader035.fdocuments.in/reader035/viewer/2022062217/5697bfa81a28abf838c994aa/html5/thumbnails/2.jpg)
Definition of a Logarithmic Function
For y > 0 and b > 0, b ≠ 1,
logb y = x if and only if bx = y
Note: Logarithmic functions are the inverse of exponential functions
Example: log2 8 = 3 since 23 = 8Read as: “log base 2 of 8”
![Page 3: 5.3 Intro to Logarithms 2/27/2013. Definition of a Logarithmic Function For y > 0 and b > 0, b ≠ 1, log b y = x if and only if b x = y Note: Logarithmic.](https://reader035.fdocuments.in/reader035/viewer/2022062217/5697bfa81a28abf838c994aa/html5/thumbnails/3.jpg)
Location of Base and Exponent in Exponential and Logarithmic Forms
Logarithmic form: x = logb y Exponential Form: bx = y
Exponent Exponent
Base Base
![Page 4: 5.3 Intro to Logarithms 2/27/2013. Definition of a Logarithmic Function For y > 0 and b > 0, b ≠ 1, log b y = x if and only if b x = y Note: Logarithmic.](https://reader035.fdocuments.in/reader035/viewer/2022062217/5697bfa81a28abf838c994aa/html5/thumbnails/4.jpg)
Basic Logarithmic Properties Involving One
• Logb b = __because 1 is the exponent to which b must be raised to obtain b. (b1 = b).
• Logb 1 = __because 0 is the exponent to which b must be raised to obtain 1. (b0 = 1).
1
0
logb y = x if and only if bx = y
![Page 5: 5.3 Intro to Logarithms 2/27/2013. Definition of a Logarithmic Function For y > 0 and b > 0, b ≠ 1, log b y = x if and only if b x = y Note: Logarithmic.](https://reader035.fdocuments.in/reader035/viewer/2022062217/5697bfa81a28abf838c994aa/html5/thumbnails/5.jpg)
Popular Bases have special names
Base 10
log 10 x = log x is called a common logarithm
Base “e”
log e x = ln x is called the natural logarithm
or “natural log”
![Page 6: 5.3 Intro to Logarithms 2/27/2013. Definition of a Logarithmic Function For y > 0 and b > 0, b ≠ 1, log b y = x if and only if b x = y Note: Logarithmic.](https://reader035.fdocuments.in/reader035/viewer/2022062217/5697bfa81a28abf838c994aa/html5/thumbnails/6.jpg)
e and Natural Logarithme is the natural base and is also called “Euler’s
number”: an irrational number (like ) and is
approximately equal to 2.718281828...Real Life Use: Compounding Interest problemRemember the formula as n approaches +
The Natural logarithm of a number x (written as “ln (x)”) is the power to which e would have to be raised to equal x.
For example, ln(7.389...) is 2, because e2=7.389Note: and ln(x) are inverse functions.
![Page 7: 5.3 Intro to Logarithms 2/27/2013. Definition of a Logarithmic Function For y > 0 and b > 0, b ≠ 1, log b y = x if and only if b x = y Note: Logarithmic.](https://reader035.fdocuments.in/reader035/viewer/2022062217/5697bfa81a28abf838c994aa/html5/thumbnails/7.jpg)
Inverse properties
Since and ln(x) are inverse functions.
and
Since are inverse functions.
and Proof: Proof: Then
![Page 8: 5.3 Intro to Logarithms 2/27/2013. Definition of a Logarithmic Function For y > 0 and b > 0, b ≠ 1, log b y = x if and only if b x = y Note: Logarithmic.](https://reader035.fdocuments.in/reader035/viewer/2022062217/5697bfa81a28abf838c994aa/html5/thumbnails/8.jpg)
Example 1 Rewrite in Exponential Form
LOGARITHMIC FORM
a. log2 16 =
EXPONENTIAL FORM
4 24 = 16
b. log7 1 = 0 70 = 1
c. log5 5 = 1 51 = 5
d. log 0.01 = 2– = 0.0110 2–
e. log1/4 4 = 1– = 44
1 – 1
logb y = x is bx = y
![Page 9: 5.3 Intro to Logarithms 2/27/2013. Definition of a Logarithmic Function For y > 0 and b > 0, b ≠ 1, log b y = x if and only if b x = y Note: Logarithmic.](https://reader035.fdocuments.in/reader035/viewer/2022062217/5697bfa81a28abf838c994aa/html5/thumbnails/9.jpg)
Example 1 Rewrite in Exponential Form
LOGARITHMIC FORM EXPONENTIAL FORM
f. ln
log e x = ln x
𝑙𝑛𝑒2=log𝑒𝑒2=2 =
g. ln ln 𝑥=log𝑒𝑥=2 =
![Page 10: 5.3 Intro to Logarithms 2/27/2013. Definition of a Logarithmic Function For y > 0 and b > 0, b ≠ 1, log b y = x if and only if b x = y Note: Logarithmic.](https://reader035.fdocuments.in/reader035/viewer/2022062217/5697bfa81a28abf838c994aa/html5/thumbnails/10.jpg)
Example 2 Rewrite in Logarithmic Form Form
LOGARITHMIC FORMEXPONENTIAL FORM
log𝟕49=2a. =
logb y = x is bx = y
log .01=−2b. =
log𝟔1=0c. =
log1
10000=− 4d. =
![Page 11: 5.3 Intro to Logarithms 2/27/2013. Definition of a Logarithmic Function For y > 0 and b > 0, b ≠ 1, log b y = x if and only if b x = y Note: Logarithmic.](https://reader035.fdocuments.in/reader035/viewer/2022062217/5697bfa81a28abf838c994aa/html5/thumbnails/11.jpg)
Example 3 Evaluate Logarithmic Expressions
Evaluate the expression.
a. log4 64 4? = 64 What power of 4 gives 64?
43 = 64 Guess, check, and revise.
log4 64 = 3
logb y = x is bx = y
41/2 = 2 Guess, check, and revise.
log4 2 =2
1
4? = 2 What power of 4 gives 2?
b. log4 2
![Page 12: 5.3 Intro to Logarithms 2/27/2013. Definition of a Logarithmic Function For y > 0 and b > 0, b ≠ 1, log b y = x if and only if b x = y Note: Logarithmic.](https://reader035.fdocuments.in/reader035/viewer/2022062217/5697bfa81a28abf838c994aa/html5/thumbnails/12.jpg)
Example 3 Evaluate Logarithmic Expressions
= 93
1 –2
Guess, check, and revise.
= 93
1What power of gives 9?
?
3
1
log1/3 9 = 2–
c. log1/3 9
Since
d.
![Page 13: 5.3 Intro to Logarithms 2/27/2013. Definition of a Logarithmic Function For y > 0 and b > 0, b ≠ 1, log b y = x if and only if b x = y Note: Logarithmic.](https://reader035.fdocuments.in/reader035/viewer/2022062217/5697bfa81a28abf838c994aa/html5/thumbnails/13.jpg)
Example 4 Simplifying Exponential Functions
a. 7 log7 5
Since = 5
b. 2log2 √3
Since =
![Page 14: 5.3 Intro to Logarithms 2/27/2013. Definition of a Logarithmic Function For y > 0 and b > 0, b ≠ 1, log b y = x if and only if b x = y Note: Logarithmic.](https://reader035.fdocuments.in/reader035/viewer/2022062217/5697bfa81a28abf838c994aa/html5/thumbnails/14.jpg)
Example 4 Simplifying Exponential Functions
c. 𝑒ln 6
Since = 6
d. 𝑒ln 𝑥3
Since =
![Page 15: 5.3 Intro to Logarithms 2/27/2013. Definition of a Logarithmic Function For y > 0 and b > 0, b ≠ 1, log b y = x if and only if b x = y Note: Logarithmic.](https://reader035.fdocuments.in/reader035/viewer/2022062217/5697bfa81a28abf838c994aa/html5/thumbnails/15.jpg)
Homework WS 5.3 odd problems only