1. 8.4 Logarithmic Functions

2. Relationship to Exponential Function
Recall the exponential
function
The inverse is
Logb x = y

3. Definition of a Logarithm with base b
Let b and y be positive numbers and b  1. The logarithm of y with base b is denoted by Logb y and is defined as
Logb y = x if and only if bx = y.
Logarithmic Form
Exponential Form
Key:
Logb y and bx = y are equivalent
The base must be positive
The number that you are taking the log of
must be positive
 The value of the log is equal to the exponent.

4. Change Logarithms to Exponential Form

5. Change Exponential Equations to Logarithmic Form And Evaluate
Example:
Evaluate Log2 64
Evaluate
Log25 5
Log6 1
Change to Exponential form

6. Common Logarithm Log10 x = y is the common logarithm.
Denoted simply as Log x
Note: If you do not see a base written with
the log, then the base is 10.

7. Natural Logarithm
Loge x = y is the natural log.
Denoted as Ln y = x.

8. Special Values of Logarithms
Logb 1 = because b0 = 1
Logb b = because b1 = b
Inverses
Logb bx = x because g(f(x)) = x
because f(g(x)) = x

9. The Graph of Logarithmic Functions y = logb (x - h) + k
x = h is the asymptote.
Domain x > h.
Range y is all real numbers.
If b > 1, the graph increases up to the right.
If 0 < b < 1, the graph reflects down. The graph decreases left to right.