1. Logarithms and Logarithmic Functions
Section 6.3 Beginning on page 310

2. Logarithms log 𝑏 𝑦 =𝑥 𝑏 𝑥 =𝑦 𝑏 𝑥 =𝑦 log 𝑏 𝑦 =𝑥
For what value of x does 2 𝑥 =6 ? Logarithms can answer this question. Log is the inverse operation to undo unknown exponents.
log 𝑏 𝑦 =𝑥
𝑏 𝑥 =𝑦
𝑏 𝑥 =𝑦
log 𝑏 𝑦 =𝑥
*Read as log base b of y
Exponential form
Logarithmic form

3. Rewriting Logarithmic Equations
Example 1: Rewrite each equation in exponential form.
log =4
log 4 1 =0
log =1
log =-1
2 4 =16
4 0 =1
12 1 =12
1 4 −1 =4
log 𝑏 𝑦 =𝑥
𝑏 𝑥 =𝑦

4. Rewriting Exponential Functions
Example 2: Rewrite each equation logarithmic form.
5 2 =25
10 −1 =0.1 =4
6 −3 = 1 216
log =2
log =−1
log 8 4 = 2 3
log =−3
𝑏 𝑥 =𝑦
log 𝑏 𝑦 =𝑥

5. Evaluating Logarithmic Expressions
Example 3: Evaluate each logarithm
a) log b) log c) log d) log 36 6
1 5 ? =125
4 ? =64
5 ? =0.2
36 ? =6
5 ? = 1 5
1 2 4 -3 −1

6. Evaluating Common and Natural Logs
A common logarithm is logarithm with base 10. It is denoted by log or simply log. A natural logarithm is a logarithm with base 𝑒 and is usually denoted by ln.
Example 4: Evaluate (a) log 8 and (b) ln 0.3 using a calculator. Round your answer to three decimal places.
a) log 8≈0.903
b) ln 0.3 ≈−1.204

7. Using Inverse Properties
Exponential functions and logarithmic functions undo each other:
log 𝑏 𝑏 𝑥 =𝑥 and 𝑏 log 𝑏 𝑥 =𝑥
Example 5: Simplify
a) 10 log b) log 𝑥 =4
= log 𝑥
= log 𝑥
=2𝑥

8. Finding Inverse Functions
Example 6: Find the inverse of each function.
a) 𝑔 𝑥 = 6 𝑥 b) 𝑦= ln (𝑥+3)
𝑥= 6 𝑦
𝑥= ln (𝑦+3)
𝑦= log 6 𝑥
𝑦+3= 𝑒 𝑥
f(𝑥)= log 6 𝑥
𝑦= 𝑒 𝑥 −3

9. Graphing Logarithmic Functions
JUST SKETCH GRAPHS

10. Graphing a Logarithmic Function
Example 7: Graph 𝑓 𝑥 = log 3 𝑥
Find the inverse function.
Create a table of values for 𝑔 𝑥
Swap the values of x and y to create a table
for 𝑓 𝑥 .
4) Plot the points and connect them with a smooth curve.
𝑥= log 3 𝑦
𝑦= 3 𝑥
𝑔(𝑥)= 3 𝑥 x -2 -1 1 2
G(x)
1 9
1 3 1 3 9 X
𝟏 𝟗
𝟏 𝟑 1 3 9
F(x) -2 -1 2

11. Monitoring Progress Rewrite the equation in exponential form.
1) log =4 2) log 7 7 =1
3) log =0 4) log =−5
Rewrite the equation in logarithmic form.
5) 7 2 =49 6) =1
7) 4 −1 = ) =2
Evaluate the logarithm. If necessary, use a calculator and round your answer to three decimal places.
9) log ) log ) log ) ln 0.75
3 4 =81
7 1 =7
1 2 −5 =32
14 0 =1
log =2
log =0
log =−1
log = 1 8
= 1 3 =5
≈1.079
≈−0.288

12. Monitoring Progress 13) 8 log 8 𝑥 14) log 7 7 −3𝑥
Simplify the expression:
13) 8 log 8 𝑥 ) log −3𝑥
15) log 𝑥 ) 𝑒 ln 20
17) Find the inverse of 𝑦= 4 𝑥
18) Find the inverse of 𝑦= ln (𝑥−5) =𝑥
=−3𝑥
=6𝑥
=20
𝑦= log 4 𝑥
𝑦= 𝑒 𝑥 +5

13. Monitoring Progress Graph the function.
19) 𝑦= log 2 𝑥 ) 𝑓 𝑥 = log 5 𝑥 ) 𝑦= log 𝑥