Chapter 11 Sec 4Chapter 11 Sec 4

Logarithmic FunctionsLogarithmic Functions

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Pre-Calculus Chapter 11 Sections 4 & 5

Graph an Exponential FunctionGraph an Exponential FunctionIf y = 2x we see exponential growth meaning as x slowly increases y grows rapidly.The inverse of this function is x = 2y this represent quantities that increase or decrease slowly.

In general the inverse of y = bx is x = by.. x = by y y is called the is called the logarithmlogarithm ofof x x and is and is usually written as usually written as yy = log = logbbx x and is read and is read

log base b of x.log base b of x.

-3 -2 -1 1 2 3 4

6

5

4

3

2

1

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Pre-Calculus Chapter 11 Sections 4 & 5

Logarithm with Base bLogarithm with Base b

NbkN kb ifonly and if log

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Pre-Calculus Chapter 11 Sections 4 & 5

Logarithmic to Exponential FormLogarithmic to Exponential Form

Write each expression in exponential form.Write each expression in exponential form.logb N = k if and only if bk = N

a. loga. log88 1 = 0 1 = 0

b = b = 88 N = N = 11 k k = 0= 0

b. logb. log55 125 = 3 125 = 3b b = 5= 5 N = N = 125 125 k k = 3 = 3

c. logc. log13 13 169 = 2169 = 2b b = 13= 13 N N = 169= 169 k k =2=2

b b = 2= 2 N = N = 1/161/16 k = k =-4-4

d. log2

1

16 4

2 4 1

16

8800 = 1 = 1

5533 = 125 = 125

131322 = 169 = 169

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Pre-Calculus Chapter 11 Sections 4 & 5

Exponential to Logarithmic FormExponential to Logarithmic Form

Write each expression in logarithmic form.Write each expression in logarithmic form.logb N = k if and only if bk = N

a. 10a. 1033 = 1000 = 1000

b = b = 1010 N = N = 10001000 k k = 3= 3

b. 3b. 333 = 27 = 27b b = 3= 3 N = N = 27 27 k k = 3 = 3

b b = 1/3= 1/3 N N = 9= 9 k k = - 2= - 2

b b = 9= 9 N = N = 33 k = k =1/21/2

log9 3 1

2

d. 91

2 3

loglog1010 1000 = 3 1000 = 3

loglog33 27 = 3 27 = 3

log 1

3

9 2

c. 1

3

2

9

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Pre-Calculus Chapter 11 Sections 4 & 5

Evaluate Logarithmic ExpressionsEvaluate Logarithmic ExpressionsEvaluate log2 64, remember logb N = k and bk = N so..find k

a. log2 64

2k = 64

2k = 26 so…

k = 6

Now, log2 64 = 6

a. log3 243

3k = 243

3k = 35 so…

k = 5

Now, log3 243 = 5

= k = k

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Pre-Calculus Chapter 11 Sections 4 & 5Evaluate Logarithmic Evaluate Logarithmic ExpressionsExpressionsEvaluate each expression. logb N = k and bk = N

a. log6 68

log6 68 = k

6k = 68

so… k = 8

log6 68 = 8

b =3 k = log3 (4x - 1)

log3 N = log3 (4x - 1)

so…

N = 4x -1

b. 3log3 4x 1

3log3 4 x 1 N

143 14log3 xx

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Pre-Calculus Chapter 11 Sections 4 & 5

PropertiesProperties

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Pre-Calculus Chapter 11 Sections 4 & 5

ExampleExample Solve each equationSolve each equation

2

164log a. 3

1

p

3

1

2

1

64p

42

1

p 2

2

2

1

4

p

16p

45log112log b. 44 xx

45112 xx5 ...315 xx

6log1loglog c. 111111 xx

6log1log 1111 xx

62 xx 062 xx

032 xx 3 2 xx X

Chapter 11 Sec 5Chapter 11 Sec 5

Common LogarithmCommon Logarithm

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Pre-Calculus Chapter 11 Sections 4 & 5

Common LogsCommon Logs• Common Logarithms Common Logarithms are all logarithms that have a are all logarithms that have a

base base of of 10…log10…log 1010 x x = log 3 = log 3• Most calculators have a key for evaluation

common logarithms.

LOG

Example 1. Use a calculator to evaluate each expression to

four decimal places.

a. log 3

b. log 0.2

LOG 3 ENTER .4771

LOG 0.2 ENTER –.6990

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Pre-Calculus Chapter 11 Sections 4 & 5

Solving Solving Solve 3x = 11

3x = 11log 3x = log

11x log 3 = log

11

Equality property

Power property

3log

11logx

1828.24771.0

0414.1x

Divide each side by log 3

Solve 5x = 62

5x = 62log 5x = log

62x log 5 = log

625log

62logx

5643.26990.

7924.1x

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Pre-Calculus Chapter 11 Sections 4 & 5

Change of Base FormulaChange of Base Formula• This allows you to write equivalent logarithmic This allows you to write equivalent logarithmic

expressions that have different bases. For expressions that have different bases. For example change base 3 into base 10example change base 3 into base 10

a

nn

b

ba log

loglog

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Pre-Calculus Chapter 11 Sections 4 & 5

Change of BaseChange of Base

Express log Express log 44 25 in terms of common logarithms. 25 in terms of common logarithms.

Then approximate its value. Then approximate its value.

4log

25log25log a.

10

104 3219.2

6021.

3980.1

3log

19log18log b.

10

103 6309.2

4771.

2553.1

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Pre-Calculus Chapter 11 Sections 4 & 5

AntilogarithmAntilogarithm• Sometime the logarithm of Sometime the logarithm of xx is know to have a is know to have a

value of value of a,a, but but xx is not known. is not known.• Then Then xx is called the is called the antilogarithm of a, antilogarithm of a, written written

as antilog as antilog a.a. • So, if log So, if log xx = = a, a, then then xx = antilog = antilog a.a.• Remember that the inverse Remember that the inverse (or antilog) (or antilog) of a of a

logarithmic function is an exponential functionlogarithmic function is an exponential function ..ie log ie log x = x = 2.7 → 2.7 → xx = antilog 2.7 or 10 = antilog 2.7 or 102.72.7

xx =501.2 =501.2

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Pre-Calculus Chapter 11 Sections 4 & 5

Daily AssignmentDaily Assignment

• Chapter 11 Sections 4 & 5Chapter 11 Sections 4 & 5• Text BookText Book

• Pgs 723 – 724 Pgs 723 – 724 • #21 – 51 Odd; #21 – 51 Odd;

• Pgs 730 – 731 Pgs 730 – 731 • #19 – 45 Odd; #19 – 45 Odd;