8.4 Logarithmic Functions

Relationship to Exponential Function

Recall the exponentialfunction

yxb

The inverse is

xb y

Logb x = y

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.

Change Logarithms to Exponential Form

Log3 9 = 2

Log5 5 = 1

Log½ 4 = -2

Log19 1 = 0

Change Exponential Equations to Logarithmic Form And Evaluate

Example:

Evaluate Log2 64Change to Exponential form

6

22

6426

x

x

x

Evaluate

Log25 5

Log6 1

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.

Natural Logarithm

Loge x = y is the natural log.

Denoted as Ln y = x.

Special Values of Logarithms

Logb 1 = 0 because b0 = 1

Logb b = 1 because b1 = b

Inverses

Logb bx = x because g(f(x)) = x

log xb bb x because f(g(x)) = x

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.