Section 3.3 Logarithmic Functionsmrsk.ca/AP/logFunctions.pdf · Logarithmic Function (Definition) For x > 0 and b > 0, b ≠ 1 g b yx is equivalent to y xb b xxb y y The function

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Section 3.3 - Logarithmic Functions

Graph: 2yx

Find the inverse function of ( ) 2xf x

2xy

2yx Solve for y?

Logarithmic Function (Definition)

For x > 0 and b > 0, b ≠ 1

logb

y x is equivalent to yx b

log b

x x b y

y

The function ( ) logbf x x is the logarithmic function with base b.

logb

x : read log base b of x log x means 10 log x

Example

Write each equation in its equivalent exponential form:

a) 7

3 log x 37x

b) 2 log 25b

225 b

c) 4

log 26 y 26 4y

Write each equation in its equivalent logarithmic form:

a) 52 x 2

5 log x

b) 327 b 3 log 27b

c) 33ye log 33e

y

Base

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Basic Logarithmic Properties

1log bb

1 bb

0log 1 0 1 b

b

Inverse Properties

log xb

b x 7

8log 7 8

log x

bb x 3

log 173 17

Example

Evaluate each expression without using a calculator:

a. 125

1log5

x35

1log5

converts to exponential

x55 3

x3

3125

1log5

5 5 5

3 1 1log log log 5 3

1 35)

25(Inverse Property

b. 73

3log

713log 7/1

3

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Natural Logarithms

Definition

( ) loge

lnf x x x

The logarithmic function with base e is called natural logarithmic function.

ln x read "el en of x"

log(-1) = doesn’t exist ln(-1) = doesn’t exist

log0 = doesn’t exist ln0 = doesn’t exist

log0.5 0.3010 ln0.5 0.6931

log1 = 0 ln1 = 0

log2 0.3010 ln 2 0.6931

log10 = 1 lne = 1

Change-of-Base Logarithmic

log

loglog a

ba

M

bM

log lnlog ln

log logb b

M Mb b

M M or

Evaluate

7

log25

7log

0625log 06

log(2506 / log(7))

4.02

Or

7

250n7ln

6llo 25 206g 4.0

ln(2506 / ln(7))

517lnlog 1.7604

ln1

57

20.1lnlog 3.3219

ln.1

20

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Domain

The domain of a logarithmic function of the form ( ) logbf x x is the set of all positive real numbers.

(Inside the log has to be > 0)

Range: ,

Example

Find the domain of

a) 4

( ) log ( 5)f x x

5 0 5x x Domain: 5,

b) ( ) ln(4 )f x x

4 - x > 0

-x > -4

x < 4 Domain: , 4

c) 2( ) ln( )h x x

x2 > 0 all real numbers except 0.

Domain: {x| x ≠ 0} or ,0 0,

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Graphs of Logarithmic Functions

Graph ( ) logg x x

Asymptote: x = 0 (Force inside log to be equal to zero, then solve for x)

( ) log5

f x x

Asymptote: x = 0

log( ) log

log55

xf x x

x g(x)

0

0.5 -.3

1 0

2 .3

3 .5

x

y

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Graph: ( ) log1/ 2

f x x

Asymptote: x = 0

Domain: 0,

Range: ,

Graph:

( ) log 12

f x x

Asymptote: x = 1

Domain: 1,

Range: ,

Shifted 1 unit right.

( ) ln( 1)f x x

Asymptote: x = 1

x

y

x

y

x

y

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Exercises Section 3.3 - Logarithmic Functions

1. Find 8

log 14

2. Write the equation in its equivalent logarithmic form: 6426

3. Write the equation in its equivalent exponential form:

x9

log2


5. Write the equation in its equivalent logarithmic form: 3 15125

6. Write the equation in its equivalent logarithmic form: 3 64 4

7. Write the equation in its equivalent logarithmic form: 3 343b

8. Write the equation in its equivalent logarithmic form: 3008 y


2723 8

10. Write the equation in its equivalent exponential form: 5

log 125 y


4

log 16 x


5

1log5

x


2

1log8

x

14. Write the equation in its equivalent exponential form

6

log 6 x

15. Write the equation in its equivalent exponential form

3

1log3

x


6 log 64


2 log x


log 81 8


1log 364

20. Evaluate the expression without using a calculator: 4

log 16


1log8


log 6

25


1log3


log 3

25. Find 5

log 8 using common logarithms

26. Find the number 5

log 1

27. Find the number 2

7log 7

28. Find the number log 8

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29. Find the number log310

30. Find the number 2 ln3e

31. Find the number 3lne

32. Find the domain of 5

log ( 4)x

33. Find the domain of

5log ( 6)x

34. Find the domain of log(2 )x

35. Find the domain of log(7 )x

36. Find the domain of 2ln( 2)x

37. Find the domain of 2ln( 7)x


2log 4 12x x


2log5

xx

40. Sketch the graph of 4

( ) log 2f x x


( ) logf x x


( ) log 2f x x

43. On a study by psychologists Bornstein and Bornstein, it was found that the average walking speed

w, in feet per second, of a person living in a city of population P, in thousands, is given by the

function:

w(P) = 0.37 ln P + 0.05

a) The population is 124,848. Find the average walking speed of people living in Hartford.

b) The population is 1,236,249. Find the average walking speed of people living in San Antonio.

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44. The loudness of sounds is measured in a unit called a decibel. To measure with this unit, we first

assign an intensity of 0

I to a very faint sound, called the threshold sound. If a particular sound has

intensity I, then the decibel rating of this louder sound is

0

10log II

d

Find the exact decibel rating of a sound with intensity

010,000I

45. Students in an accounting class took a final exam and then took equivalent forms of the exam at

monthly intervals thereafter. The average score S(t), as a percent, after t months was found to be

given by the function

S(t) = 78 – 15 log(t + 1), t 0

a) What was the average score when the students initially took the test, t = 0?

b) What was the average score after 4 months? 24 months?

46. A model for advertising response is given by the function

N(a) = 1000 + 200 ln a, a 1

Where N(a) is the number of units sold when a is the amount spent on advertising, in thousands of

dollars.

a) N(a = 1)

b) N(a = 5)

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Solution Section 3.3 - Logarithmic Functions

Exercise

Find 8

log 14

Solution

8

log14log 14

log8

1.2691

Exercise

Write the equation in its equivalent logarithmic form 6426

Solution

26 log 64

Exercise

Write the equation in its equivalent exponential form

x9

log2

Solution

x 29

Exercise


Solution

625log4 5

Exercise


15 3

Solution

1251

5log3

24

Exercise


Solution

464 3/1

31log

64

Exercise

Write the equation in its equivalent logarithmic form 3433 b

Solution

3343log b

Exercise

Write the equation in its equivalent logarithmic form 3008 y

Solution

y 300 8

log

Exercise

Write the equation in its equivalent logarithmic form: 3

2723 8

Solution

23

27log 38

Exercise


y125log5

Solution

1255 y

Exercise


4

log 16 x

Solution

16 4x

25

Exercise


5

1log5

x

Solution

1 55

x

Exercise


2

1log8

x

Solution

31 22

x

32 2x

Exercise


6

log 6 x

Solution

1/26 6x

Exercise


3

1log3

x

Solution

1/23 3x

Exercise

Write the equation in its equivalent exponential form: 2

6 log 64

Solution

2

664 2log 66 4

Exercise

Write the equation in its equivalent exponential form: 9

2 log x

Solution

9

92 log 2x x

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Exercise

Write the equation in its equivalent exponential form: 881log3

Solution

8

3log 81 8 81 3

Exercise

Write the equation in its equivalent exponential form: 3641log

4

Solution

431log 3 1

6464x

Exercise

Evaluate the expression without using a calculator: 4

log 16

Solution

4 4

2log 16 log 4 2

logb

xb x

Exercise


1log8

Solution

2 2 31 1log log8 2

2

3log 2

logb

xb x

3

Exercise


log 6

Solution

6 6

1/2log 6 log 6 1 2

27

Exercise


1log3

Solution

3 3 1/21 1log log3 3

3

1/2log 3

logb

xb x

12

Exercise


log 3

Solution

x 7/1

33log

Converts to exponential

x33 7/1

71x

713log 7

3

Exercise

Find 5

log 8 using common logarithms

Solution

ln 1.292ln

log 5 5

88

Exercise

Find the number 5

log 1

Solution

5log 1 0

28

Exercise

Find the number 2

7log 7

Solution

2

7log 7 2

Exercise

Find the number log 8

33

Solution

log 833 8

Exercise

Find the number log310

Solution

log310 3

Exercise

Find the number 2 ln3e

Solution

2 ln3 22.1672e

Exercise

Find the number 3lne

Solution

3ln 3e

Exercise

Find the domain of 5

log ( 4)x

Solution

4 4,x

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Exercise

Find the domain of 5

log ( 6)x

Solution

6 6,x

Exercise

Find the domain of )2log( x

Solution

2 0x

2x

2 ,2x

Exercise

Find the domain of )7log( x

Solution

7 0x

7x

7 ,7x

Exercise

Find the domain of 2)2ln( x

Solution

2 0 2x x

,2 2,

Exercise

Find the domain of 2)7ln( x

Solution

7 0 7x x

,7 7,

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Exercise

Find the domain of 2log 4 12x x

Solution

2 4 12 0 2,6x x x

, 2 2,6 6,

Exercise

Find the domain of 2log5

xx

Solution

2

5

x

x

, 5 2,

Exercise

Sketch the graph of 4

( ) log 2f x x

Solution

Asymptote: x = 2

Domain: 2,

Range: ,

-5 0 2

+ - +

x f(x)

2

2.5 .5

3 0

4 .5

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Exercise


( ) logf x x

Solution

Asymptote: x = 0

Domain: , 0 0,

Range: ,

Exercise


( ) log 2f x x

Solution

Asymptote: x = 0

Domain: 0,

Range: ,

x f(x)

0

.5 .5

1 0

2 .5

x f(x)

0

0.5 2.5

1 0

2 1.5

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Exercise

On a study by psychologists Bornstein and Bornstein, it was found that the average walking speed w, in

feet per second, of a person living in a city of population P, in thousands, is given by the function

w(P) = 0.37 ln P + 0.05

a) The population is 124,848. Find the average walking speed of people living in Hartford.

b) The population is 1,236,249. Find the average walking speed of people living in San Antonio.

Solution

124,848 = 124.848 thousand

a) w(P=124.848) = 0.37 ln(124.848) + 0.05 1.8 ft/sec

b) w(P=1,236.249) = 0.37 ln(1,236.249) + 0.05 2.7 ft/sec

Exercise

The loudness of sounds is measured in a unit called a decibel. To measure with this unit, we first assign

an intensity of 0

I to a very faint sound, called the threshold sound. If a particular sound has intensity I,

then the decibel rating of this louder sound is

0

10log II

d

Find the exact decibel rating of a sound with intensity 0

10,000I

Solution

0

0

1000010log

I

Id

10log10000

40

Exercise

A model for advertising response is given by the function

N(a) = 1000 + 200 ln a, a 1

Where N(a) is the number of units sold when a is the amount spent on advertising, in thousands of dollars.

a) N(a = 1)

b) N(a = 5)

Solution

a) N(a=1) = 1000 + 200 ln1 = 1000 units

b) N(a=5) = 1000 + 200 ln5 = 1322 units

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Exercise

Students in an accounting class took a final exam and then took equivalent forms of the exam at monthly

intervals thereafter. The average score S(t), as a percent, after t months was found to be given by the

function

S(t) = 78 – 15 log(t + 1), t 0



Solution


t = 0 → S(t) = 78 – 15 log(0 + 1) = 78%


After 4 months → S(t = 4) = 78 – 15 log(4 + 1) = 67.5%

24 months → S(t = 24) = 78 – 15 log(24 + 1) = 57%

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Exercise

3log125

15125

15

3

312log 28

83/1

0.301010 2 3010.02log

xQt txQ

log

3679.01 e 13679.0 ln

3kp kp

3log

4log 7t

74 t

log7 0.845 701 .8450

9676.038.0 ln 38.0e .96760

5lnW t 5e Wt

log8 = 0.9031

ln(-4) doesn’t exist

ln(0.00037) = -7.9020

Documents

Section 3.3 Logarithmic Functionsmrsk.ca/AP/logFunctions.pdf · Logarithmic Function (Definition) For x > 0 and b > 0, b ≠ 1 g b yx is equivalent to y xb b xxb y y The function