4.3 - Logarithmic Functions

Section 4.3 Logarithmic Functions

Chapter 4 – Exponential and Logarithmic Functions

4.3 - Logarithmic Functions

Exponential Functions

Recall from last class that every exponential function f (x) = ax with a >0 and a 1 is a one-to-one function and therefore has an inverse function.

That inverse function is called the logarithmic function with base a and is denoted by loga.

4.3 - Logarithmic Functions

DefinitionLogarithmic Function

Let a be a positive number with a 1. The logarithmic function with base a, denoted by loga, is defined by

logax = y ay = x

So logax is the exponent to which the base a must be raised to give x.

4.3 - Logarithmic Functions

Switching Between Logs & Exp.

NOTE: logax is an exponent!

4.3 - Logarithmic Functions

Example – pg. 322 #5

Complete the table by expressing the logarithmic equation in exponential form or by expressing the exponential equation into logarithmic form.

4.3 - Logarithmic Functions

Properties of Logarithms

4.3 - Logarithmic Functions

Example – pg. 322Use the definition of the logarithmic function to find

x.

2 229. a) log 5 b) log 16

1 136. a) log 6 b) log 3

2 3x x

x x

4.3 - Logarithmic Functions

Graphs of Logarithmic Functions

Because the exponential and logarithmic functions are inverses with each other, we can learn about the logarithmic function from the exponential function. Remember,

Characteristic Exponential Logarithmic

Domain (-∞, ∞)

Range (0, ∞)

x-intercept None

y-intercept (0,1)

VA None

HA y = 0

4.3 - Logarithmic Functions

Graphs of Log Functions

4.3 - Logarithmic Functions

Example – pg. 323Graph the function, not by plotting points or using a

graphing calculator, but by starting from the graph of a logax function. State the domain, range, and asymptote.

53.

58.

2log 4f x x

3log 1 2y x

4.3 - Logarithmic Functions

DefinitionsCommon Logarithm

The logarithm with base 10 is called the common logarithm and is denoted by omitting the base:

Natural LogarithmThe logarithm with base e is called the natural logarithm and is denoted by:

10log logx x

ln logex x

4.3 - Logarithmic Functions

NoteBoth the common and natural logs can be evaluated

on your calculator.

4.3 - Logarithmic Functions

Properties of Natural Logs

4.3 - Logarithmic Functions

Example – pg. 322Find the domain of the function.

266. ( ) ln

67. = ln ln 2

g x x x

h x x x