1. Objectives & Vocabulary
Write equivalent forms for exponential and logarithmic functions.
Write, evaluate, and graph logarithmic functions.
logarithm common logarithm
logarithmic function

2. Notes #1-2 Rewrite in other form
Exponential Equation
Logarithmic Form
92= 81
33 = 27
x0 = 1(x ≠ 0)
Logarithmic Form
Exponential Equation
log = 3
log12144 = 2
log 8 = –3 a. b. c.
3A. Change 64 = 1296 to logarithmic form
B. Change log279 = to exponential form. 2 3
Calculate (without a calculator). 1 27
4. log864
5. log3

3. 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

4. Example 1: 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

5. Example 2: 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
Notes #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. Write each logarithmic form in exponential equation.
Notes #2
Write each logarithmic form in exponential equation.
Logarithmic Form
Exponential Equation
log = 3
log12144 = 2
log 8 = –3
The base of the logarithm becomes the base of the power.
103 = 1000
122 = 144
The logarithm is the exponent. 1 2 –3
= 8
An logarithm can be negative. 1 2

8. 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.

9. A logarithm with base 10 is called a common logarithm
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.

10. Example 3A: Evaluating Logarithms by Using Mental Math
Evaluate by without a calculator.
log 0.01
10? = 0.01
The log is the exponent.
10–2 = 0.01
Think: What power of 10 is 0.01?
log 0.01 = –2

11. Example 3B: Evaluating Logarithms by Using Mental Math
Evaluate without a calculator.
log5 125
5? = 125
The log is the exponent.
log5125 = 3

12. Example 3C/3D: Evaluating Logarithms by Using Mental Math
Evaluate without a calculator. 1 5
3c. log5
log = –1 1 5
3d. log250.04
log = –1

13. 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.
The domain of y = 2x is all real numbers (R), and the range is {y|y > 0}. The domain of y = log2x is {x|x > 0}, and the range is all real numbers (R).

14. Example 4A: Graphing Logarithmic Functions
Use the x-values {–2, –1, 0, 1, 2}. Graph the function and its inverse. Describe the domain and range of the inverse function.
f(x) = x 1 2
Graph f(x) = x by using a table of values. 1 2 x –2 –1 1 2
f(x) =( ) x 4 1 2 4

15. To graph the inverse, f–1(x) = log x, by using a table of values.
Example 4A Continued
To graph the inverse, f–1(x) = log x, by using a table of values. 1 2 x 4 2 1
f –1(x) =log x –2 –1 1 2 4
The domain of f–1(x) is {x|x > 0}, and the range is R.

16. 3A. Change 64 = 1296 to logarithmic form log61296 = 4
Notes (continued)
3A. Change 64 = 1296 to logarithmic form
log61296 = 4
27 = 9 2 3
B. Change log279 = to exponential form. 2 3
Calculate the following using mental math (without a calculator).
4. log864 2
5. log3 1 27 –3

17. Notes (graphing)
6. Use the x-values {–1, 0, 1, 2} to graph
f(x) = 3x Then graph its inverse. Describe the domain and range of the inverse function.
D: {x > 0}; R: all real numbers