Graphs of Logarithmic Functions - Purdue University

16-week Lesson 31 (8-week Lesson 25)

Graphs of Logarithmic Functions

Example 1: Complete the input/output table for the function

?(? ) = log 2 (? ), and use the ordered pairs to sketch the graph of the

function. After graphing, list the domain, range, zeros, positive/negative

intervals, increasing/decreasing intervals, and the intercepts.

Inputs

Outputs

(these are

powers of

the base 2)

(these are exponents that

produce the powers of the

base 2)

?(? ) = log 2 (? )

?(? ) ¡ú

?¡ú0

?¡Þ

1

1

log 2 ( ) = ??

16

16

1

1

log 2 ( ) = ??

8

8

1

1

log 2 ( ) = ??

4

4

1

1

log 2 ( ) = ??

2

2

?(?)

Outputs

?

Notice that the graph of

?(?) = log 2 (?) is

increasing throughout its

domain. When a function

is always increasing or

always decreasing that

function is one-to-one, and

it will have an inverse

function. The inverse of a

logarithmic function is an

exponential function.

Domain: (?, ¡Þ)

Range: (?¡Þ, ¡Þ)

1

log 2 (1) = ?

2

log 2 (2) = ?

Positive intervals:

?(?) > 0 when ? is (?, ¡Þ)

4

log 2 (4) = ?

Negative intervals:

?(?) < 0 when ? is (?, ?)

8

log 2 (8) = ?

?????????? ?????????:

?(?) ?? ?????? ???? ? ?? (?, ¡Þ)

16

log 2 (16) = ?

?¡ú¡Þ

? (? ) ¡ú ¡Þ

Zeros:

?(?) = 0 when ? = ?

Decreasing intervals:

?(?) is falling when ? is ????

Intercepts:

? ? intercept: (?, ?)

? ? intercept: ????

1

16-week Lesson 31 (8-week Lesson 25)

Graphs of Logarithmic Functions

Vertical Asymptote:

- a vertical line (? = #) that the graph of a function approaches, but

never touches or crosses, when the inputs approach an undefined

value (? ¡ú #, where # is a value that is not part of the domain)

o in the case of ? (? ) = log 2 (? ), as the inputs get closer and closer

to zero (? ¡ú 0), the outputs get smaller and smaller

(?(? ) ¡ú ?¡Þ), so the graph has a vertical asymptote at ? = 0

o the graph of every logarithmic function will have a vertical

asymptote (? = #)

- in a later example (Example 2) I will denoted the vertical asymptote

with a dotted line to make it easier to identify

Example 2: Re-write the function ?(? ) = log 2 (? + 2) in terms of

?(? ) = log 2 (? ). Then find the ?-intercept of ? and find its graph by

transforming the graph of the original function ?. Enter exact answers

only (no approximations) for the ?-intercept.

?(?)

Outputs

?(?) = log 2 (?)

Re ? write ?(? )

in terms of ? (? ):

?(?) =

?

Inputs

When solving problems like this on

the homework, you can use the

transformation and the ?-intercept

to get the graph, or you can simply

use transformations only to get the

graph, and then identify the

?-intercept from the graph.

? ? ?????????:

0 = log 2 (? + 2)

20 = ? + 2

1=?+2

?1 = ?

(??, ?)

To find the ?-intercept algebraically, remember that a logarithm

is an exponent, so anything equal to a logarithm is also an

exponent. In this example, the logarithm log 2 (? + 2) represents

the exponent that makes the base 2 equal to the argument ? + 2.

Since 0 is equal to log 2 (? + 2), 0 is the exponent that makes 2

equal to ? + 2.

2

16-week Lesson 31 (8-week Lesson 25)

Graphs of Logarithmic Functions

Example 3: Re-write each of the following functions in terms of

?(? ) = log 2 (? ), then match the transformation with the appropriate

graph. Also, find the ?-intercepts of each function.

?(?)

Outputs

a. ?(? ) = ? log 2 (? )

?(?) = log 2 (?)

?(?) =

? ? ?????????:

?

Inputs

0 = ? log 2 (? ) ¡ú

(

,0)

?(?)

Outputs

b. ?(? ) = log 2 (?? )

?(?) = log 2 (?)

?(?) =

? ? ?????????:

?

Inputs

0 = log 2 (?? ) ¡ú

(

,0)

? ? ?????????:

3

16-week Lesson 31 (8-week Lesson 25)

Graphs of Logarithmic Functions

LON-CAPA Problem :

Given the function ? (? ) =

complete the following:

, along with it¡¯s graph below,

a. Express the new function ?(? ) = 2

the original function ?.

?+2

in terms of

?(?) =

? ? ?????????:

?+2

b. Find the ?-intercept of the function ?(? ) = 2

and enter

your answer as an ordered pair (?, ?). Enter exact answers only, no

approximations.

? ? ????????? (?, ?) =

c. Transform the graph of ? to get the graph of ?(? ) = 2?+2

. Use

the ?-intercept of ? to verify that your transformation is correct.

?(?)

Outputs

Keep in mind that even

though the point you¡¯re

given on the graph maybe

the ?-intercept, after

transforming the point it

may no longer be an

intercept. Use the answer

from part a. to determine

how to transform the

point that you are given

on the graph. After

transforming the graph,

use your answer from part

b. to verify that the

?-intercept on the new

graph is correct.

?

Inputs

4

16-week Lesson 31 (8-week Lesson 25)

Graphs of Logarithmic Functions

Example 4: Re-write each of the following functions in terms of

?(? ) = log 2 (? ), then match the transformation with the appropriate

graph. Also, find the ?-intercepts of each function.

?(?)

Outputs

a. ?(? ) = (log 2 (? )) ? 2

?(?) = log 2 (?)

?(?) =

? ? ?????????:

?

Inputs

0 = (log 2 (? )) ? 2 ¡ú

(

,0)

? ? ?????????:

?(?) = log 2 (?)

b. ?(? ) = log 2 (2? )

?(?) =

?

Inputs

? ? ?????????:

0 = log 2 (2? ) ¡ú

(

,0)

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