1. Objectives
Write equivalent forms for exponential and logarithmic functions.
Write, evaluate, and graph logarithmic functions.

2. How many times would you have to double $1 before you had $8?
You could use an exponential equation to model this situation. 1(2x) = 8.
You may be able to solve this equation by using mental math if you know 23 = 8.
So you would have to double the dollar 3 times to have $8.

3. How many times would you have to double $1 before you had $512?
You could solve this problem if you could solve 2x = 8 by using an inverse operation that undoes raising a base to an exponent equation to model this situation.
This operation is called finding the logarithm.
A logarithm is the exponent to which a specified base is raised to obtain a given value.

4. You can write an exponential equation as a logarithmic equation and vice versa.
Read logb a= x, as “the log base b of a is x.” Notice that the log is the exponent.
Reading Math

5. Example 1: Converting from Exponential to Logarithmic Form
Write each exponential equation in logarithmic form.
Exponential Equation
Logarithmic Form
35 = 243
25 = 5
104 = 10,000
6–1 =
ab = c
The base of the exponent becomes the base of the logarithm.
log3243 = 5 1 2 1 2
log255 =
The exponent is the logarithm.
log1010,000 = 4 1 6
log = –1 1 6
An exponent (or log) can be negative.
logac =b
The log (and the exponent) can be a variable.

6. Write each exponential equation in logarithmic form. 92= 81 33 = 27
Check It Out! Example 1
Write each exponential equation in logarithmic form.
Exponential Equation
Logarithmic Form
92= 81
33 = 27
x0 = 1(x ≠ 0)
The base of the exponent becomes the base of the logarithm. a.
log981 = 2 b.
log327 = 3
The exponent of the logarithm.
The log (and the exponent) can be a variable. c.
logx1 = 0

7. Example 2: Converting from Logarithmic to Exponential Form
Write each logarithmic form in exponential equation.
Logarithmic Form
Exponential Equation
log99 = 1
log = 9
log82 =
log = –2
logb1 = 0
The base of the logarithm becomes the base of the power.
91 = 9
The logarithm is the exponent.
29 = 512 1 3 1 3
8 = 2
A logarithm can be a negative number. 1 1 16
4–2 = 16
Any nonzero base to the zero power is 1.
b0 = 1

8. Write each logarithmic form in exponential equation.
Check It Out! Example 2
Write each logarithmic form in exponential equation.
Logarithmic Form
Exponential Equation
log1010 = 1
log12144 = 2
log 8 = –3
The base of the logarithm becomes the base of the power.
101 = 10
122 = 144
The logarithm is the exponent. 1 2 –3
= 8
An logarithm can be negative. 1 2

9. A logarithm is an exponent, so the rules for exponents also apply to logarithms. You may have noticed the following properties in the last example.

10. A logarithm with base 10 is called a…. common logarithm.
If no base is written for a logarithm, the base is assumed to be 10. For example, log 5 = log105.
You can use mental math to evaluate some logarithms.

11. Example 3A: Evaluating Logarithms by Using Mental Math
Evaluate by using mental math.
log 0.01 -2
= ________
10x = 0.01
The log is the exponent.
10–2 = 0.01
Think: What power of 10 is 0.01?
log 0.01 = –2

12. Example 3B: Evaluating Logarithms by Using Mental Math
Evaluate by using mental math.
log5 125 3
= ________
5x = 125
The log is the exponent.
53 = 125
Think: What power of 5 is 125?
log5125 = 3

13. Example 3C: Evaluating Logarithms by Using Mental Math
Evaluate by using mental math. 1 5
log5 -1
= ________
5x = 1 5
The log is the exponent.
5–1 = 1 5 1 5
Think: What power of 5 is ?
log = –1 1 5

14. -5 Check It Out! Example 3a Evaluate by using mental math. log 0.00001
= ________
10x =
The log is the exponent.
10–5 =
Think: What power of 10 is 0.01?
log = –5

15. -1 Check It Out! Example 3b Evaluate by using mental math. log250.04
= ________
25x = 0.04
The log is the exponent.
25–1 = 0.04
Think: What power of 25 is 0.04?
log = –1

16. Exponential and logarithmic operations undo each other since they are inverse operations.

17. Example 4: Recognizing Inverses
Simplify each expression.
a. log3311
b. log381
c. 5log510
log3311
log334
5log510 10 4 11

18. log 100.9 2log2(8x) 0.9 8x Check It Out! Example 4
a. Simplify log100.9
b. Simplify 2log2(8x)
log 100.9
2log2(8x)
0.9 8x

19. Because logarithms are the inverses of exponents, the inverse of an exponential function, such as
y = 2x, is a logarithmic function, such as y = log2x.
You may notice that the domain and range of each function are switched.
Domain
Range

20. 1. Change 64 = 1296 to logarithmic form. log61296 = 4
Lesson Quiz: Part I
1. Change 64 = 1296 to logarithmic form.
log61296 = 4
27 = 9 2 3
2. Change log279 = to exponential form. 2 3
Calculate the following using mental math.
3. log 100,000 5
4. log648
0.5
5. log3 1 27 –3

21. Lesson Quiz: Part II
6. Use the x-values {–2, –1, 0, 1, 2, 3} to graph f(x) =( )X. Then graph its inverse. Describe the domain and range of the inverse function. 5 4
- 2 -1 1 2 3
- 2
.64 -1 .8 1
1.25 2
1.5625 3
1.9531
D: {x > 0};
R: all real numbers