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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.
5.1 Exponential Functions
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
5.1 Exponential Functions
Graphs of exponential functions
Important Characteristics
One-to-one functionDomain:Y-intercept (0,1)Range:
Rx
,0y
5.1 Exponential Functions
10, bandbbxf x
Increasing if b>1 Decreasing if 0<b<1
5.1 Exponential Functions
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
5.1 Exponential Functions
EXPONENTIAL EQUATIONS WITH LIKE BASESTHE UNIQUENESS PROPERTY
23 12525 xx
2332 55
xx
636 55 xx
636 xx
2x
5.1 Exponential Functions
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
5.2 Logarithms and Logarithmic Functions
Graphing Logarithmic FunctionsCalculators and Common Logarithms
5.2 Logarithms and Logarithmic Functions
Pg 493 #87 and 88Earthquake Intensity
5.2 Logarithms and Logarithmic Functions
Homework pg 491 1-94
5.3 The Exponential Function and Natural Logarithms
Natural Logarithmic Function
5.3 The Exponential Function and Natural Logarithms
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
5.3 The Exponential Function and Natural Logarithms
Using Properties of Logarithms
3
2
lnn
m 4log yx
5.3 The Exponential Function and Natural Logarithms
7log28log 33
Using Properties of Logarithms
15
25 log2log xxx
5.3 The Exponential Function and Natural Logarithms
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
5.3 The Exponential Function and Natural Logarithms
152log5 008.0log 2.0
Using the change of base formula
5.3 The Exponential Function and Natural Logarithms
Homework pg 502 1-106
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
5.4 Exponential/Logarithmic Equations and Applications
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
5.4 Exponential/Logarithmic Equations and Applications
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
,
5.4 Exponential/Logarithmic Equations and Applications
Solving Exponential Equations
753 1 xe
123 1 xe
41 xe
4lnln 1 xe
4ln1x
14ln x
5.4 Exponential/Logarithmic Equations and Applications
Solving Exponential Equations
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
5.4 Exponential/Logarithmic Equations and Applications
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
5.4 Exponential/Logarithmic Equations and Applications
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
5.4 Exponential/Logarithmic Equations and Applications
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
5.4 Exponential/Logarithmic Equations and Applications
Homework pg 516 1-106
Chapter 5 Review
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