Logarithmic Functions

Definition of a Logarithmic Function

For x > 0 and b > 0, b = 1,

y = logb x is equivalent to by = x.

The function f (x) = logb x is the logarithmic function with base b.

Location of Base and Exponent in Exponential and Logarithmic Forms

Logarithmic form: y = logb x Exponential Form: by = x. Logarithmic form: y = logb x Exponential Form: by = x.

Exponent Exponent

Base Base

Text Example

Write each equation in its equivalent exponential form.a. 2 = log5 x b. 3 = logb 64 c. log3 7 = y

Solution With the fact that y = logb x means by = x,

c. log3 7 = y or y = log3 7 means 3y = 7.

a. 2 = log5 x means 52 = x.

Logarithms are exponents.Logarithms are exponents.

b. 3 = logb 64 means b3 = 64.

Logarithms are exponents.Logarithms are exponents.

Evaluatea. log2 16 b. log3 9 c. log25 5

Solution

log25 5 = 1/2 because 251/2 = 5.25 to what power is 5?c. log25 5

log3 9 = 2 because 32 = 9.3 to what power is 9?b. log3 9

log2 16 = 4 because 24 = 16.2 to what power is 16?a. log2 16

Logarithmic Expression Evaluated

Question Needed for Evaluation

Logarithmic Expression

Text Example

Basic Logarithmic Properties Involving One

• Logb b = 1 because 1 is the exponent to which b must be

raised to obtain b. (b1 = b).

• Logb 1 = 0 because 0 is the exponent to which b must be

raised to obtain 1. (b0 = 1).

Inverse Properties of Logarithms

For x > 0 and b 1,• logb bx = xThe logarithm with base b

of b raised to a power equals that power.

• b logb x = x b raised to the logarithm with base b of a number equals that number.

Properties of Common Logarithms

General Properties Common Logarithms

1. logb 1 = 0 1. log 1 = 0

2. logb b = 1 2. log 10 = 1

3. logb bx = 0 3. log 10x = x4. b logb x = x 4. 10 log x = x

Examples of Logarithmic Properties

log 4 4 = 1

log 8 1 = 0

3 log 3 6 = 6

log 5 5 3 = 3

2 log 2 7 = 7

Properties of Natural Logarithms

General Properties Natural Logarithms

1. logb 1 = 0 1. ln 1 = 0

2. logb b = 1 2. ln e = 1

3. logb bx = 0 3. ln ex = x4. b logb x = x 4. e ln x = x

Examples of Natural Logarithmic Properties

e log e 6 = e ln 6 = 6

log e e 3 = 3

Problems

Use the inverse properties to simplify:27 ln 4

ln

7.1 log

1. ln 2.

3. 4. log1000

5. log10 6. 10

x x

x

e

e e

e

Characteristics of the Graphs of Logarithmic Functions of the Form f(x) = logbx

• The x-intercept is 1. There is no y-intercept.• The y-axis is a vertical asymptote. (x = 0)• If 0 < b < 1, the function is decreasing. If b > 1, the function is

increasing. • The graph is smooth and continuous. It has no sharp corners or

edges.

-2 -1

6

2 3 4 5

5

4

3

2

-1-2

6

f (x) = logb xb>1

-2 -1

6

2 3 4 5

5

4

3

2

-1-2

6

f (x) = logb x0<b<1

Graph f (x) = 2x and g(x) = log2 x in the same rectangular coordinate system.Solution We first set up a table of coordinates for f (x) = 2x. Reversing these coordinates gives the coordinates for the inverse function, g(x) = log2 x.

4

2

8211/21/4f (x) = 2x

310-1-2x

2

4

310-1-2g(x) = log2 x

8211/21/4x

Reverse coordinates.

Text Example

Solution

We now sketch the basic exponential graph. The graph of the inverse (logarithmic) can also be drawn by reflecting the graph of f (x) = 2x over the line y = x.

-2 -1

6

2 3 4 5

5

4

3

2

-1-2

6

f (x) = 2x

f (x) = log2 x

y = x

Text ExampleGraph f (x) = 2x and g(x) = log2 x in the same rectangular coordinate system.

Examples

( ) log( 1)

( ) log( 1)

( ) 1 ln

f x x

g x x

h x x

Graph using transformations.

Domain of Logarithmic Functions

Because the logarithmic function is the inverse of the exponential function, its domain and range are the reversed.

The domain is { x | x > 0 } and the range will be all real numbers.

For variations of the basic graph, say the domain will consist of all x for which x + c > 0.

Find the domain of the following:

1.

2.

3.

( ) logbf x x c

( ) log 3g x x

4( ) log 5h x x

2( ) ln 3f x x

Sample Problems

Find the domain of

4( ) log ( 3)

( ) ln(2 1)

( ) log(4 )

f x x

g x x

h x x