Logarithmic Functions

Lesson 8.4

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.

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”.

Example 1: Rewriting Logarithmic Equations

Logarithmic FormA) log3 81 = 4

B) log4 1 = 0

C) log9 9 = 1

D) log

E) log3 3 = 1

F) log2 .125 = -3

Exponential Form

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

Example 2: Evaluating Logarithmic Expressions

A) log2 64 =

B) log

C) log

D) log4 2 =

E) Log5 125 =

F) Log8 2 =

Example 3: Using Inverse Properties

A) 10log 2.3

B) Log2 8x

C) 10log x

D) Log3 81x

Example 4: Finding Inverses

A) y = log

B) y = ln (x – 2)

C) y = log2 x

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.

Example 5: Graphing Logarithmic Functions

A) y = log B) y = log2 (x + 1) + 2