Chapter 4
Exponential and
Logarithmic
Functions
4.1 Exponential Functions
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Objectives:
•
•
•
•
Evaluate exponential functions.
Graph exponential functions.
Evaluate functions with base e.
Use compound interest formulas.
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Definition of the Exponential Function
The exponential function f with base b is defined by
x
y

b
f ( x)  b x
or
where b is a positive constant other than 1 (b > 0 and
b  1) and x is any real number.
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Example: Evaluating an Exponential Function
The exponential function f ( x)  42.2(1.56) x models the
average amount spent, f(x), in dollars, at a shopping
mall after x hours. What is the average amount spent, to
the nearest dollar, after three hours at a shopping mall?
We substitute 3 for x and evaluate the function.
f ( x)  42.2(1.56) x
f (3)  42.2(1.56)3  160.20876  160
After 3 hours at a shopping mall, the average amount
spent is $160.
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Example: Graphing an Exponential Function
Graph: f ( x)  3
x
–2
–1
x
We set up a table of coordinates,
then plot these points, connecting
x
f ( x)  3
them with a smooth, continuous
1 curve.
2
f (2)  3 
9
1
1
f (1)  3 
3
0
f (0)  30  1
1
f (1)  31  3
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Example: Transformations Involving Exponential
Functions
x 1
x
g
(
x
)

3
f
(
x
)

3
Use the graph of
to obtain the graph of
x
f
(
x
)

3
Begin with
We’ve identified
three points and the
asymptote.
 1, 1 


 3
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(1,3)
(0,1)
Horizontal
asymptote
y=0
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Example: Transformations Involving Exponential
Functions (continued)
x 1
x
g
(
x
)

3
f
(
x
)

3
Use the graph of
to obtain the graph of
(1,3)
The graph will shift
1 unit to the right.
Add 1 to each
x-coordinate.
(0,1)
 1, 1 


 3
 0, 1 


 3
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(2,3)
(1,1)
Horizontal
asymptote
y=0
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Characteristics of Exponential Functions of the Form
f ( x)  b x
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The Natural Base e
The number e is defined as the value that 1  1 


 n
n
approaches as n gets larger and larger. As n  
the approximate value of e to nine decimal places is
e  2.718281827
The irrational number, e, approximately 2.72, is called
the natural base. The function f ( x)  e x is called the
natural exponential function.
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Example: Evaluating Functions with Base e
The exponential function f ( x)  1066e0.042 x models the
gray wolf population of the Western Great Lakes, f(x), x
years after 1978. Project the gray wolf’s population in
the recovery area in 2012.
Because 2012 is 34 years after 1978, we substitute 34
for x in the given function.
f ( x)  1066e0.042 x
f (34)  1066e0.042(34)  4446
This indicates that the gray wolf population in the
Western Great Lakes in the year 2012 is projected to
be approximately 4446.
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Formulas for Compound Interest
After t years, the balance, A, in an account with
principal P and annual interest rate r (in decimal form)
is given by the following formulas:
1. For n compounding periods per year:
nt
r

A  P 1  
 n
2. For continuous compounding:
A  Pert
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Example: Using Compound Interest Formulas
A sum of $10,000 is invested at an annual rate of 8%.
Find the balance in the account after 5 years subject to
quarterly compounding.
We will use the formula for n compounding periods per
year, with n = 4.
nt
4 5
r
0.08 

A  P 1  
A  10,000 1 
 14,859.47

 n
4 

The balance of the account after 5 years subject to
quarterly compounding will be $14,859.47.
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Example: Using Compound Interest Formulas
A sum of $10,000 is invested at an annual rate of 8%.
Find the balance in the account after 5 years subject to
continuous compounding.
We will use the formula for continuous compounding.
A  Pe
rt
A  10,000e
0.08(5)
 14,918.25
The balance in the account after 5 years subject to
continuous compounding will be $14,918.25.
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