Chapter 4 Exponential and Logarithm Functions

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Chapter 4Exponential and Logarithm Functions
4.1 Exponential Functions 4.2 The Natural
Exponential Function 4.3 Logarithm
Functions 4.4 Logarithmic Transformations 4.5
Logistic Growth
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Section 4.1Exponential Functions
Review of Laws of Exponents (p157
158) Characteristics of Exponential
Functions Modeling with Exponential Functions
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Basic Exponential Functions
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Basic Exponential Functions f(x) 4x
x -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0
y4x
f(x) 4x domain reals range positive reals y
intercept 1 increasing a gt 1
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Basic Exponential Functions f(x) 4x
x -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0
y4x
as x increases without bound, 4x increases
without bound as x decreases without bound, 4x
gets close to 0
as x ? 8, 4x ? 8 as x ? -8, 4x ? 0 y 0 is a
Horizontal Asymptote
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Basic Exponential Functions f(x) (¼)x
x -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0
y(¼)x
f(x) (¼)x domain reals range positive
reals y intercept 1 decreasing 0 lt a lt 1
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Basic Exponential Functions f(x) (¼)x
x -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0
y(¼)x
as x increases without bound, (¼)x gets close
to 0. as x decreases without bound, (¼)x
increases without bound
as x ? 8, (¼)x ? 0 y 0 is a Horizontal
Asymptote as x ? -8, (¼)x ? 8
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Basic Exponential Functions
f(x) 4x and g(x) (¼)x graphs are reflections
about the y axis Does f(-x) g(x)?
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Basic Exponential Functions
f(x) (a)x and
f(x) cax
f(x) 54x domain reals range positive reals
HA y 0 y intercept 5
f(x) 4x domain reals range positive reals
HA y 0 y intercept 1
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Basic Exponential Functions f(x)
cax Characteristics
f(x) cax domain reals range positive
reals HA y 0 y intercept c increasing for a
gt 1decreasing for 0 lt a lt 1
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Variations of Basic Exponential Functions
f(x) 54x domain reals range positive reals
HA y 0 y intercept 5
f(x) 20 54x domain reals range reals gt
20 HA y 20 y intercept 25
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Modeling with Exponential Functions
Example/156 A math student pours himself a mug
of steaming coffee and then forgets to drink it.
In a room that remains at 20C, the coffee cools,
losing heat rapidly at first and then more slowly
as the liquid approaches room temperature. The
coffee is initially 90C and after 10 minutes
cools to 68C.
Find a model for the temperature of the coffee
over time.
exponential with 0 lt a lt 1 and shifted up 20
units H(t) V cat
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Modeling with Exponential Functions
H(t) V cat
Can we solve for V, c, and a?
V 20 so H(t) 20 cat
Vertical shift
H(0) 9020 ca0 9020c 90c 70 so
H(t) 20 70at
Initial Temperature
H(10) 6820 70a10 6870a10 48a10
48/70a (48/70)(1/10)a 0.963 so H(t) 20
70(0.963)t
Another data point
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Modeling with Exponential Functions
Example/156 A gymnastics team practices its
balance-beam routine, improving month by month.
Initially the average score is 3.8 but then
scores increase rapidly as more time passes,
additional efforts result in small gains. In
fact, after 6 months of practice, the average
score is 5.7
Find a model for the average score of the team
over time.
exponential with 0 lt a lt 1, reflected about x
axis and shifted up. S(t) V - cat
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Modeling with Exponential Functions
S(t) V - cat
Can we solve for V, c, and a?
V 10 so S(t) 10 - cat
Vertical shift
S(0) 3.810 - ca0 3.810 - c 3.8c 6.2
so S(t) 10 6.2at
Initial Score
S(6) 5.710 6.2a6 5.7-6.2a6 -4.3a6
-4.3/-6.2a (4.3/6.2)(1/6)a 0.94 so S(t)
10 6.2(0.94)t
Another data point
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Section 4.2The Natural Exponential Function
What is e? Base-e Exponential Functions
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Natural Exponential Function f(x) ex
(2.718)x
x -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0
yex
f(x) ex domain reals range positive reals y
intercept 1 increasing a gt 1
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Homework Pages 193-194 1-16 Turn In 7,8, 13