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Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.
Exponential and Logarithmic Functions
5• Exponential Functions
• Logarithmic Functions
• Compound Interest
• Differentiation of Exponential Functions
• Differentiation of Logarithmic Functions
• Exponential Functions as Mathematical Models
Copyright © 2006 Brooks/Cole, a division of Thomson Learning, 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
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 the Exponential Function
,
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, 1xy f 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
The logarithm of x to the base b is defined by
Ex.
43
07
-2
1/ 3
15
log 81 4; 3 81
log 1 0; 7 1
1log 9 2; 81
3
log 5 1; 5 5
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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
Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.
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
The logarithmic function of x to the base b is defined by
Properties:
1. Domain: (0, )2. Range:
3. x-intercept: (1, 0)4. Continuous on (0, )5. Increasing on (0, ) if b > 1
Decreasing on (0, ) if b < 1
,
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Graphs of Logarithmic Functions
Ex. 3( ) logf x x
(1,0)
3xy
3logy x
1
3
x
y
1/ 3logy x
1/ 3( ) logf x x
x x
yy
(1,0)
(0, 1) (0, 1)
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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.
Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.
ExampleA normal child’s systolic blood pressure may be approximated by the function
where p(x) is measured in millimeters of mercury, x is measured in pounds, and m and b are constants. Given that m = 19.4 and b = 18, determine the systolic blood pressure of a child who weighs 92 lb.
( ) (ln )p x m x b
Since 19.4, 92, and 18
we have (92) 19.4(ln 92) 18
105.72
m x b
p
Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.
Compound Interest Formula
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
Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.
ExampleHow long will it take an investment of $10,000 to grow to $15,000 if it earns an interest at the rate of 12% / year compounded quarterly?
40.12
1 15,000 10,000 14
mt tr
A Pm
4 41.5 1.03 ln(1.03) ln1.5
4 ln1.0
Taking ln both sides
log l3 ln1.5
ln1.5 3.43 years
4ln1.0
og
3
nb
t t
bm n mt
t
Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.
Effective Rate of Interest
eff 1 1m
rr
m
reff = Effective rate of interest r = Nominal interest rate/year m = 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
It is about 6.7% per year.
Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.
Present Value Formula for Compound Interest
1mt
rP A
m
A = The accumulated amount after mt periods
P = Principalr = Nominal interest rate/yearm = Number of periods/yeart = Number of years
Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.
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
Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.
Continuous Compound Interest Formula
rtA Pe
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
Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.
Differentiation of Exponential Functions
x xde e
dx
( ) ( ) ( )f x f xde e f x
dx
Chain Rule for Exponential Functions
Derivative of Exponential Function
If f (x) is a differentiable function, then
Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.
ExamplesFind the derivative of
3 5( ) .xf x e
3 4 4 4( ) 4 4x xf x x e x e 3 44 1xx e x
Find the relative extrema of 4 4( ) .xf x x e
3 5( ) 3 5x df x e x
dx
3 55 xe
-1 0
+ – +
Relative Min. f (0) = 0Relative Max. f (-1) = 4
1
e
f
x
Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.
Differentiation of Logarithmic Functions
1ln 0
dx x
dx x
( )ln ( ) ( ) 0
( )
d f xf x f x
dx f x
Chain Rule for Exponential Functions
Derivative of Exponential Function
If f (x) is a differentiable function, then
Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.
ExamplesFind the derivative of 2( ) ln 2 1 .f x x
1( ) 2f x
x
Find an equation of the tangent line to the graph of ( ) 2 ln at 1, 2 .f x x x
2
2
2 1( )
2 1
dx
dxf xx
2
4
2 1
x
x
(1) 3f
2 3( 1)
3 1
y x
y x
Slope:
Equation:
Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.
Logarithmic Differentiation
1. Take the Natural Logarithm on both sides of the equation and use the properties of logarithms to write as a sum of simpler terms.
2. Differentiate both sides of the equation with respect to x.
3. Solve the resulting equation for .dy
dx
Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.
ExamplesUse logarithmic differentiation to find the derivative of 5
3 2 9 1y x x
5ln ln 3 2 9 1y x x
5ln ln 3 2 ln 9 1y x x
1ln ln 3 2 5ln 9 1
2y x x
1 3 5(9)
2 3 2 9 1
dy
y dx x x
5 3 45
3 2 9 12 3 2 9 1
dyx x
dx x x
Apply ln
Differentiate
Properties of ln
Solve
Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.
Exponential Growth/Decay Models
0( ) 0ktQ t Q e t
Q0 is the initial quantity
k is the growth/decay constant
A quantity Q whose rate of growth/decay at any time t is directly proportional to the amount present at time t can be modeled by:
Growth
Decay 0( ) 0ktQ t Q e t
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ExampleA certain bacteria culture experiences exponential growth. If the bacteria numbered 20 originally and after 4 hours there were 120, find the number of bacteria present after 6 hours.
0 20 soQ ( ) 20 ktQ t e4120 20 ke
ln 60.4479
4k
0.4479(6)(6) 20 294Q e
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Learning Curves
( ) 0ktQ t C Ae t
C, A, k are positive constants
An exponential function may be applied to certain types of learning processes with the model:
0,C A
y = C
x
y
Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.
ExampleSuppose that the temperature T, in degrees Fahrenheit, of an object after t minutes can be modeled using the following equation: 0.3( ) 200 150 tT t e
1. Find the temperature of the object after 5 minutes.0.3(5)(5) 200 150 166.5T e
2. Find the time it takes for the temperature of the object to reach 190°.
0.3190 200 150 te 0.31/15 te ln 1/15
9 min.0.3
t
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Logistic Growth Model
A, B, k are positive constants
An exponential function may be applied to a logistic growth model:
( ) 01 kt
AQ t t
Be
y = A
0,1
A
B
x
y
Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.
Example
2400( )
1 1199 ktR t
e
The number of people R, in a small school district who have heard a particular rumor after t days can be modeled by:
If 10 people know the rumor after 1 day, find the number who heard it after 6 days.
(1)
240010
1 1199 ke
1 1199 240ke 239
ln 1.6131199
k …
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1.613
2400( )
1 1199 tR t
e
So
1.613(6)
2400(6) 2232 people
1 1199R
e
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Example
5000( )
1 1249 ktQ t
e
The number of soldiers at Fort MacArthur who contracted influenza after t days during a flu epidemic is approximated by the exponential model:
If 40 soldiers contracted the flu by day 7, find how many soldiers contracted the flu by day 15.
(7)
500040
1 1249 ke
71 1249 125ke 1 124
ln 0.337 1249
k …
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