Math-3

Lesson 5-2

The

Logarithm Function

Finding the Inverse: exchange the locations

of ‘x’ and ‘y’ in the equation then solve for ‘y’.

2)2()( xxf

2)2( xy

2)2( yx

2)2( yx

2 yx

yx 2

xy 2

Is the Inverse of a function a function?

-2

-1

0

1

2

3

4

5

6

0 2 4 6 8 10

Horizontal line test: if a horizontal line passes

through the graph of the relation at more than one

location, then the inverse of that relation is NOT

a function.

0

1

2

3

4

5

6

7

8

9

10

-2 -1 0 1 2 3 4 5 6

2)2( xy

Fails horizontal line test

xy 22)2( yx or

The inverse relation

is not a function.

Domain, Range, and Inverse Functions

Domain: The input values (that have

corresponding outputs)

Range: The output values (that

have corresponding inputs)

Inverse of a Function:A function resulting from an

“exchange” of the inputs

and outputs.

RangeDomainxf ,:)(

:)(1 xf Domain = range of f(x)

Range = domain of f(x)

xxf log)(

Exponential Function

xxf 10)(

Domain = ?

Range = ?

Domain = ?

Range = ?

(-∞, ∞)

(0, ∞)

Logarithm

Function

Inverse

Functions

(0, ∞)

(-∞, ∞)

Horizontal

asymptote = ?y = 0 Vertical

asymptote = ?x = 0

Transformations of the Log Function

xxf log)( )1log()( xxgRight 1 shift

Domain = ?

Range = ?

(1, ∞)

(-∞, ∞)vertical asymptote = ?x = 0

asymptote = ? x = 1X-intercept = ?

x = 1

Domain = ? (0, ∞)

Range = ? (-∞, ∞)

Where increasing = ?

(0, ∞)

Logarand

5)12log(3)( xxg

Logarand

Vertical Asymptote: The value of ‘x’ that makes

the logarand equal to zero.

Vertical asymptote = ? 2x – 1 = 0

x = ½

Evaluating Logs on your calculator

?8log

log 8) =

Push buttons:

0.903089987

?10ln

Push buttons:

ln 10) = 2.302585093

?0log

?)3log(

error

error

Transformations of the Log Function

xxf log)(

3)1log(2)( xxg

VSF = 2

left 1 translation

Down 3 translation

Domain = ?

Range = ?

(-1, ∞)

(-∞, ∞)

asymptote = ? x = -1

Transformations of the Log Function

xxf log)(

1)2log(3)( xxg

Reflected x-axis

VSF = 3

Right 2 translation

Up 1 translation

Domain = ?

Range = ?

(2, ∞)

(-∞, ∞)

asymptote = ? x = 2

NOT exponential (has a

vertical asymptote NOT

a horizontal asymptote.

What is a logarithm?

A logarithm is another way of writing an exponent.

x8log282 x

Both of these equations are saying

the same thing:

“2 raised to what power is 8?”

x is the exponent Log = exponent

Exponential LogarithmForm Form

823 38log2

base base

“base 2 to the 3rd

power is 8”“log base 2 of 8 is 3”

x9log393 x

3 to what power is 9 ? 3 to what power is 9?

x = 2

Log =

Exponential LogarithmForm Form

x25log5

x64log4

x81log9

x1000log10

255 x

644 x

ybx xyb logWhy did they

use “b”?

819 x

100010 x

Log =

Your Turn: Log =

1. 366 x

Convert to logarithm form

2. 15 x

3. 162 x

4. x52

5. x43

Your Turn: Log =

6.

Convert to exponential form

7.

8.

9.

10.

x100log10

x27log3

x1log9

2log4 x

5log2 x

Vocabulary

Common Logarithm: has a base of 10.

x100log10

We usually write it in this form: x100log

Natural Logarithm: has a base of e.

We always write it in this form:

1718.2log e

1718.2ln

Graphs of the Common and Natural Logarithm

Base increasing:

Base increasing:

Your Turn:

What is the base?

11.

12.

13.

x5ln

x20log

x8log2

Evaluating Logs on your calculator

?8log

log 8) =

Push buttons:

0.903089987

?10ln

Push buttons:

ln 10) = 2.302585093

Estimate the value of the log:

?8log x8log 810 x

010 x10

1

110

108

8.0x

9.0x

Find on your calculator.8log 903.08log

Estimate the value of the log:

?50log x50log 5010 x

110 x10

10

210

10050

5.1x

6.1x

Find on your calculator.50log 7.150log

Your Turn:

Estimate the value of the log:

14.

15.

16.

?10log

?5ln

?8log x8log 810 x

Finding the Inverse

xxf )5(2)(

?)(1 xf

yx )5(2

yx)5(

2 Base: 5

“A log is an exponent”

2log5

xy

2log)( 5

1 xxf

Finding the Inverse

2)3()( 1 xxf

?)(1 xf

Base: 3“A log is an exponent”

2log1 3 xy

12log)( 3

1 xxf

2)3( 1 yx

1)3(2 yx

“isolate” the exponential”

12log3 xy

Finding the Inverse

2)3()( 1 xxf

?)(1 xf

12log)( 3

1 xxf

Right 1 up 2 Right 2 up 1

Finding the Inverse

)1(log2)( 2 xxf

?)(1 xf

Base: 2

“A log is an exponent”

221x

y

)1(log2 2 yx

)1(log2

2 yx

“Isolate the log”

12 2 x

y

12)( 21 x

xf