5.5 Logarithmic Functions

Objective To Define and apply logarithms

Logarithmic Functions

x = 2y is an exponential equation.

Its inverse (solving for y) is called a logarithmic equation.

Let’s look at the parts of each type of equation:

Exponential Equationx = a y

exponent

base

number

/logarithm

y = loga xLogarithmic Equation

if and only if xy alog yax

Example 1: Rewrite in exponential form and

solve loga64 = 2

a2 = 64

a = 8

Example: Solve log5 x = 3

Rewrite in exponential form:

53 = x

x = 125

base number exponent

Example 2: Solve

7y = 1/49

y = –2

log7

1

49y

An equation in the form y = logb x where b > 0 and b 1 is called a logarithmic function.

Logarithmic and exponential functions are inverses of each other

logb y = x, y = bx, logb bx = x

by = x, y = logb x, blogb x = x

Example 3. Evaluate each:a. log8 8

4

b. 6[log6 (3y – 1)]

logb bx = x

log8 84 = 4

blogb x = x

6[log6 (3y – 1)] = 3y – 1Here are some special logarithm values:

1. loga 1 = 0 because a0 = 1

2. loga a = 1 because a1 = a

3. loga ax = x because ax = ax

Example 4 : Find

5log 25 5log 1252

1log

82

55 25 log 25 2

355 25 log 125 3

32

1 12 log 3

8 8

The logarithm with base 10 is called the common logarithmic (this is the one your calculator evaluates with the log key).

To use a calculator to evaluate logarithms with other bases, you can change the base to 10 by using the following formula:

Change of Base Formula: For all positive numbers a, b, and n, where a ≠ 1 and b ≠ 1,

Example: Approximate log4 22

≈ 2.2295

a

nn

b

ba log

loglog

4log

22log22log4

6021.0

3424.1

 

Example 5. Two loud stereos are playing the same music simultaneously at 80 dB each. What is the decibel level of the combined sound? By how many decibels is the decibel level of the two stereos greater than the decibel level of one stereo?

The logarithm with base e is called the natural logarithmic (this is the one your calculator evaluates with the ln key).

To use a calculator to evaluate logarithms with other bases, you can change the base to e by using the following formula:

Change of Base Formula: For all positive numbers a, b, and n, where a ≠ 1 and b ≠ 1,

Example: Approximate log3 50

≈ 3.56088

a

nn

b

ba log

loglog

3ln

50ln50log3

0986.1

912.3

The base b logarithmic function is the inverse of the base b exponential function.

Domain of xf x b All reals

Range of xf x b Positive reals

Domain of 1 logbf x x Positive reals

Range of 1 logbf x x All reals

The most important logarithmic function in advanced mathematics and statistics has the number e as its base.

The natural logarithm of x is usually denoted ln x although sometimes it is written loge x

if and only if  

ln x k ke x

For example ln 5 1.6 because e 1.6 = 5

Example 6. Find the value of x to the nearest hundredth.

10 75x 75xe

How do you graph a logarithmic function?

Example 7: Graph f(x) = log3 x

This is the inverse of g(x) = 3x

We will need to create a table of values.(Keep in mind that logarithmic functions are inverses of exponential functions)

x g(x)

-2-1 0 1 2

1/91/3 1 3 9

x f(x)

-2-1 0 1 2

1/91/3 1 3 9

f(x) = log3 x

g(x) = 3x

Assignment

P. 194 #2 – 18 (even), 35 – 49 (odd)