Lesson 5-6: Logarithms
and Logarithmic
Functions
Advanced Math Topics
Key Concept

Logarithms with base b
log b x  y

Read: “Log of x base b is y”
Key Concepts

Common Logarithm: log10 can just be written
log
•
•

Log 100 is the same as log10 100
There is a button on your calculator for log you can only
use it when you are dealing with base 10!
Natural Logarithm: loge is called the natural log
and is written ln
•
•
Loge1 is the same as the ln 1
There is a button on your calculator for ln you can use
this anytime you see ln in a problem!
Logarithmic to Exponential
Form
1. log6 216 = 3
2. log4 1 = 0
3. log3 81 = 4
4. ln e2 = 2
5. log 40 = 1.6
Exponential to Logarithmic
Form
6. y = log10 100
8. log5 625
7. log2 0.25 = y
9. log10 0.001
Write each equation in
logarithmic form
10. 29 = 512
12. e5 =148.4
11. 0.04-3/2 = 125
Evaluate Logarithmic Expressions
13. log 43
14. log 4300
15. ln 0.05
16. log2 32
17. log5 1/125
18. log11 1/121
Solve
19. logx 25 = 2
20. logx e = 1/2
20. logx 9/100 = -2
Characteristics of Logarithmic
Functions






1. Inverse of the exponential function
y=bx
2.Continous and one-to-one
3. Domain is all positive real numbers
4. y-axis is an asymptote
5. Range is ARN
6. Contains (1,0), so x-intercept is 1
Helpful Hint

Since exponential and logarithmic functions
are inverses if the bases are the same they
“undo” each other…
log 6 68  8
3log3 ( 4 x 1)  4 x  1
Logarithmic Equations

Property of Equality
•
If b is a positive number other than 1, then
log b x  log b y if and only if x = y.
log 7 x  log 7 3
x3
Example 4

Solve
5
log 4 n 
2
Example 5

Solve log 9 x 
3
2
Example 6

Solve log x  5
16
2
Example 7

Solve
4
log 8 x 
3
Example 8

Solve log 5 ( p 2  2)  log 5 p
Example 9

2
Solve log 3 ( x  15)  log 3 2 x
Example 10

2
Solve log 14 (m  30)  log 14 m
Example 11

Solve log 4 x 2  log 4 (4 x  3)