1. Logarithmic Functions
Lesson 8.4

2. Vocabulary
Common Logarithm: the logarithm with base 10. It is denoted by log10 or simply by log.
Natural Logarithm: the logarithm with base e. It can be denoted by loge but it is more often denoted by ln.

3. Definition of Logarithm with Base b
Let b and y be positive numbers , b ≠ 1. The logarithm of y with base b is denoted by logb y and is defined as follows: logb y = x if and only if bx = y The expression logb y is read as “log base b of y”.

4. Example 1: Rewriting Logarithmic Equations
Logarithmic Form
log3 81 = 4
log4 1 = 0
log9 9 = 1
log
log3 3 = 1
log = -3
Exponential Form

5. Special Logarithmic Values
Let b be a positive real number such that b ≠ 1.
Logarithm of 1 :
logb 1 = 0 because b0 = 1
Logarithm of base b :
logb b = 1 because b1 = b

6. Example 2: Evaluating Logarithmic Expressions

7. Example 3: Using Inverse Properties
10log 2.3
Log2 8x
10log x
Log3 81x

8. Example 4: Finding Inverses
y = log
y = ln (x – 2)
y = log2 x

9. Graphs of Logarithmic Functions
The graph of y = logb (x – h) + k has these characteristics:
The line x = h is a vertical asymptote.
The domain is x > h, and the range is all real numbers
If b > 1 the graph moves up to the right. If 0<b<1, the graph moves down to the right.

10. Example 5: Graphing Logarithmic Functions
A) y = log
B) y = log2 (x + 1) + 2