Chapter 5 Exponential and Logarithmic Functions. 5.1 Exponential Functions Exponential Functions For...

Chapter 5Exponential

and Logarithmic Functions

5.1 Exponential Functions

Exponential Functions

For b > 0, b≠1, f(x) = bx defines the base b exponential function.

The domain of f is all real numbers.


Exponential Properties

Given a, b, x, and t are real numbers, with b, c > 0,

txtx bbb txt

x

bb

b xttx bb

xxx cbbc xx

bb

1

xx

b

a

a

b


Graphs of exponential functions

Important Characteristics

One-to-one functionDomain:Y-intercept (0,1)Range:

Rx

,0y


10, bandbbxf x

Increasing if b>1 Decreasing if 0<b<1


EXPONENTIAL EQUATIONS WITH LIKE BASESTHE UNIQUENESS PROPERTY

If bm = bn, then m = n.

If m = n, then bm = bn.

813 12 x

412 33 x

412 x

2

5x


EXPONENTIAL EQUATIONS WITH LIKE BASESTHE UNIQUENESS PROPERTY

23 12525 xx

2332 55

xx

636 55 xx

636 xx

2x


Homework pg 482 1-68

5.2 Logarithms and Logarithmic Functions

Logarithmic Functions

For b > 0, b ≠ 1, the base-b logarithmic function is defined as

yb bxxy ifonly and if log

Write in exponential form

8log3 2 1log0 2

823 120

Write in logarithmic form

2

12 1

2792

3

2

1log1 2 27log

2

39


Graphing Logarithmic FunctionsCalculators and Common Logarithms


Pg 493 #87 and 88Earthquake Intensity

5.3 The Exponential Function and Natural Logarithms

Natural Logarithmic Function


Properties of LogarithmsGiven M, N, and b are positive real numbers, where b ≠ 1, and any real number x.

Product Property:

“the log of a product is equal to a sum of logarithms”

Quotient Property:

“The log of a quotient is equal to a difference of logarithms”

Power Property:

“The log of a number to a power is equal to the power times the log of the number”

NMMN bbb logloglog

NMN

Mbbb logloglog

MxM bx

b loglog


Using Properties of Logarithms

3

2

lnn

m 4log yx


7log28log 33

Using Properties of Logarithms

15

25 log2log xxx


Change of Base Formula

Given the positive real numbers M, b, and d, where b≠1 and d≠1,

ebasebase

b

MM

b

MM bb

10

ln

lnlog

log

loglog


152log5 008.0log 2.0

Using the change of base formula

5.4 Exponential/Logarithmic Equations and Applications

Writing Logarithmic and Exponential Equations in Simplified Form

43loglog 22 xx

43log2 xx

43log 22 xx

1lnln2ln xxx

1ln2ln

x

xx

xx

x2ln

1ln0

1

2

1ln0

x

x

x

1

2ln0

2

x

x


Writing Logarithmic and Exponential Equations in Simplified Form

1225325400 21.0 xe

900400 21.0 xe

25.221.0 xe

xxx eee 231

xx ee 214

12

14

x

x

e

e

1214 xxe

112 xe


Solving Exponential Equations

For any real numbers b, x, and k, where b>0 and b≠1

kx

k

kifx

x

10

1010

log

log10log

,10

kx

ke

keifx

x

ln

lnln

,

b

kx

kbx

kbif x

log

log

loglog

,



753 1 xe

123 1 xe

41 xe

4lnln 1 xe

4ln1x

14ln x



192201

258009.0

te

te 009.0201192258

te 009.0201192

258

te 009.0201192

258

te 009.0

20

1192258

te 009.0ln20

1192258

ln

t009.020

1192258

ln

t

009.0

20

1192258

ln


Solving Logarithmic Equations

For real numbers b, m, and n where b > 0 and b≠1,

nmthen

nmif bb

loglog

nmthen

nmif

bb loglog

Equal bases imply equal arguments


Solving Logarithmic Equations

9loglog12log xxx

9log12

log

xx

x

912

xx

x

xxx 912 2

1280 2 xx

Use quadratic formula to solve for x


An advertising agency determines the number of items sold is related to the amount spent on advertising by the equation N(A)= 1500 + 315 ln A, where A represents the advertising budget and N(A) gives the number of sales. If a company wants to generate 5000 sales, how much money should be set aside for advertising? Round interest to the nearest dollar.

AAN ln3151500

Aln31515005000

Aln3153500

Aln315

3500

Aee ln315

3500

Ae 315

3500

Chapter 5 Review

Chapter 5 Exponential and Logarithmic Functions. 5.1 Exponential Functions Exponential Functions For...

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