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Exponential and Logarithmic Functions
5• Exponential Functions
• Logarithmic Functions
• Compound Interest
• Models
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Inc.
Exponential Function
( ) 0, 1xf x b b b
An exponential function with base b and exponent x is defined by
Ex. ( ) 3xf x Domain: All reals
Range: y > 0(0,1)
( )y f x
0 1
1 3 2 9
11 3
x y
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Laws of ExponentsLaw Example
1. x y x yb b b
2.x
x yy
bb
b
4.x x xab a b
3.yx xyb b
5.x x
x
a a
b b
1/ 2 5 / 2 6 / 2 32 2 2 2 8 12
12 3 93
55 5
5
61/ 3 6 / 3 2 18 8 8
64
3 3 3 32 2 8m m m 1/ 3 1/ 3
1/ 3
8 8 2
27 327
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Properties of Exponential Functions
,
1. The domain is .
2. The range is (0, ).
3. It passes through (0, 1).
4. It is continuous everywhere.
5. If b > 1 it is increasing on . If b < 1 it is decreasing on .
( ) 0, 1xf x b b b
,
,
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ExamplesEx. Simplify the expression.
Ex. Solve the equation
42 1/ 2
3 7
3x y
x y
4 8 2
3 7
3 x y
x y
5
5
81x
y
3 1 4 24 2x x 2 3 1 4 22 2x x
6 2 4 22 2x x 6 2 4 2x x
2 4x 2x
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Logarithms
log if and only if 0yby x x b x
An logarithmic of x to the base b is defined by
Ex. 3
7
1/ 3
5
log 81 4
log 1 0
log 9 2
log 5 1
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ExamplesEx. Solve each equation
a.
b.
2log 5x 52 32x
27log 3 x3 27x
33 3 x1 3x1
3x
m na a m n
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Notation: Common LogarithmNatural Logarithm
10log log
ln loge
x x
x x
Laws of Logarithms1. log log log
2. log log log
3. log log
4. log 1 0
5. log 1
b b b
b b b
nb b
b
b
mn m n
mm n
n
m n m
b
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ExampleUse the laws of logarithms to simplify the expression:
7 1/ 25 5 5 5log 25 log log logx y z
7
525
logx y
z
5 5 51
2 7 log log log2
x y z
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Logarithmic Function
( ) log 0, 1bf x x b b
An logarithmic function of x to the base b is defined by
Properties
1. Domain: (0, )2. Range:
3. Intercept: (1, 0)4. Continuous on (0, )5. Increasing on (0, ) if b > 1
Decreasing on (0, ) if b < 1
,
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Logarithmic Function Graphs
Ex. 3( ) logf x x
(1,0)
3xy
3logy x
1
3
x
y
1/ 3logy x
1/ 3( ) logf x x
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ln 0
ln for any real number
x
x
e x x
e x x
and lnxe x
Ex. Solve 2 1110
3xe
2 1 30xe 2 1 ln(30)x
ln(30) 11.2
2x
Apply ln to both sides.
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Compound Interest
1mt
rA P
m
A = the accumulated amount after mt periods
P = Principalr = Nominal interest rate per yearm = Number of periods/yeart = Number of years
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ExampleFind the accumulated amount of money after 5 years if $4300 is invested at 6% per year compounded quarterly.
1mt
rA P
m
4(5).06
4300 14
A
= $5791.48
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Effective Rate of Interest
eff 1 1m
rr
m
reff = Effective rate of interestr = Nominal interest rate/yearm = number of conversion periods/year
Ex. Find the effective rate of interest corresponding to a nominal rate of 6.5% per year, compounded monthly.
eff 1 1m
rr
m
12.065
1 112
.06697
So about 6.67% per year.
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Present Value for Compound Interest
1mt
rP A
m
A = the accumulated amount after mt periods
P = Principalr = Nominal interest rate per yearm = Number of periods/yeart = Number of years
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ExampleFind the present value of $4800 due in 6 years at an interest rate of 9% per year compounded monthly.
1mt
rP A
m
12(6).09
4800 112
P
$2802.83
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Inc.
Continuous Compound Interest
P rtA e
A = the accumulated amount after t years
P = Principalr = Nominal interest rate per yeart = Number of years
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ExampleFind the accumulated amount of money after 25 years if $7500 is invested at 12% per year compounded continuously.
rtA Pe0.12(25)7500A e
$150,641.53