Exponential and Logarithmic Functions Functions

INTRODUCTORY MATHEMATICAL ANALYSIS
For Business, Economics, and the Life and Social Sciences
2011 Pearson Education, Inc.
Chapter 4
Exponential and Logarithmic Functions

• To introduce exponential functions and their
applications.
• To introduce logarithmic functions and their
graphs.
• To study the basic properties of logarithmic
functions.
• To develop techniques for solving logarithmic
and exponential equations.
Chapter 4: Exponential and Logarithmic Functions
Chapter Objectives

Exponential Functions
Logarithmic Functions
Properties of Logarithms
Logarithmic and Exponential Equations
4.1)
4.2)
4.3)
4.4)
Chapter Outline

• The function f defined by
where b > 0, b  1, and the exponent x is any real
number, is called an exponential function with
base b1
.
4.1 Exponential Functions
  x
b
x
f 

The number of bacteria present in a culture after t
minutes is given by .
a. How many bacteria are present initially?
b. Approximately how many bacteria are present
after 3 minutes?
Solution:
a. When t = 0,
b. When t = 3,
Example 1 – Bacteria Growth
 
4
300
3
t
N t
 
  
 
0
4
(0) 300 300(1) 300
3
N
 
  
 
 
3
4 64 6400
(3) 300 300 711
3 27 9
N
   
   
   
   

Graph the exponential function f(x) = (1/2)x
.
Solution:
Example 3 – Graphing Exponential Functions with 0 < b < 1

Properties of Exponential Functions

Solution:
Compound Interest
• The compound amount S of the principal P at the end of n
years at the rate of r compounded annually is given by
.
Example 5 – Graph of a Function with a Constant Base
2
Graph 3 .
x
y 
(1 )n
S P r
 

Example 7 – Population Growth
The population of a town of 10,000 grows at the rate
of 2% per year. Find the population three years from
now.
Solution:
For t = 3, we have .
3
(3) 10,000(1.02) 10,612
P  

Example 9 – Population Growth
The projected population P of a city is given by
where t is the number of years after
1990. Predict the population for the year 2010.
Solution:
For t = 20,
0.05(20) 1
100,000 100,000 100,000 271,828
P e e e
   
0.05
100,000 t
P e


Example 11 – Radioactive Decay
A radioactive element decays such that after t days
the number of milligrams present is given by
.
a. How many milligrams are initially present?
Solution: For t = 0, .
b. How many milligrams are present after 10 days?
Solution: For t = 10, .
0.062
100 t
N e

 
mg
100
100 0
062
.
0

 
e
N
 
mg
8
.
53
100 10
062
.
0

 
e
N

4.2 Logarithmic Functions
Example 1 – Converting from Exponential to Logarithmic Form
• y = logbx if and only if by
=x.
• Fundamental equations are and
logb x
b x

log x
b b x

2
5
4
a. Since 5 25 it follows that log 25 2
b. Since 3 81 it follo
Exponential Form Logarithmic Form
 
 3
0
10
ws that log 81 4
c. Since 10 1 it follows that log 1 0

 

Example 3 – Graph of a Logarithmic Function with b > 1
Sketch the graph of y = log2x.
Solution:

Example 5 – Finding Logarithms
a. Find log 100.
b. Find ln 1.
c. Find log 0.1.
d. Find ln e-1
.
d. Find log366.
  2
10
log
100
log
2


0
1
ln 
1
10
log
1
.
0
log 1


 
1
ln
1
ln 1





e
e
2
1
6
log
2
6
log
6
log36 


Example 7 – Finding Half-Life
• If a radioactive element has decay constant λ, the
half-life of the element is given by
A 10-milligram sample of radioactive polonium 210
(which is denoted 210
Po) decays according to the
equation. Determine the half-life of 210
Po.
Solution:

2
ln

T
days
λ
T 4
.
138
00501
.
0
2
ln
2
ln




4.3 Properties of Logarithms
Example 1 – Finding Logarithms
• Properties of logarithms are:
n
m
mn b
b
b log
log
)
(
log
.
1 

n
m
n
m
b
b log
log
log
.
2 b 

m
r
m b
r
b log
log
3. 
a.
b.
c.
d.
7482
.
1
8451
.
0
9031
.
0
7
log
8
log
)
7
8
log(
56
log 






6532
.
0
3010
.
0
9542
.
0
2
log
9
log
2
9
log 




8062
.
1
)
9031
.
0
(
2
8
log
2
8
log
64
log 2




3495
.
0
)
6990
.
0
(
2
1
5
log
2
1
5
log
5
log 2
/
1




b
m
m
b
m
m
a
a
b
b
b
b
b
log
log
log
.
7
1
log
.
6
0
1
log
.
5
log
1
log
4.






Example 3 – Writing Logarithms in Terms of Simpler
Logarithms
a.
b.
w
z
x
w
z
x
zw
x
zw
x
ln
ln
ln
)
ln
(ln
ln
)
ln(
ln
ln








)]
3
ln(
)
2
ln(
8
ln
5
[
3
1
)]
3
ln(
)
2
ln(
[ln
3
1
)}
3
ln(
]
)
2
(
{ln[
3
1
3
)
2
(
ln
3
1
3
)
2
(
ln
3
)
2
(
ln
8
5
8
5
8
5
3
/
1
8
5
3
8
5




























x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x

Example 5 – Simplifying Logarithmic Expressions
a.
b.
c.
d.
e.
.
3
ln 3
x
e x

3
3
0
10
log
0
1000
log
1
log 3






9
8
9
/
8
7
9 8
7 7
log
7
log 

1
)
3
(
log
3
3
log
81
27
log 1
3
4
3
3
3 
















 
0
)
1
(
1
10
log
ln
10
1
log
ln 1






 
e
e

Example 7 – Evaluating a Logarithm Base 5
Find log52.
Solution:
4307
.
0
5
log
2
log
2
log
5
log
2
log
5
log
2
5





x
x
x
x

4.4 Logarithmic and Exponential Equations
• A logarithmic equation involves the logarithm of
an expression containing an unknown.
• An exponential equation has the unknown
appearing in an exponent.

An experiment was conducted with a particular type
of small animal. The logarithm of the amount of
oxygen consumed per hour was determined for a
number of the animals and was plotted against the
logarithms of the weights of the animals. It was found
that
where y is the number of microliters of oxygen
consumed per hour and x is the weight of the animal
(in grams). Solve for y.
Example 1 – Oxygen Composition
x
y log
885
.
0
934
.
5
log
log 


Solution:
Example 1 – Oxygen Composition
)
934
.
5
log(
log
log
934
.
5
log
log
885
.
0
934
.
5
log
log
885
.
0
885
.
0
x
y
x
x
y





885
.
0
934
.
5 x
y 

Example 3 – Using Logarithms to Solve an Exponential
Equation
Solution:
.
12
4
)
3
(
5 1

 
x
Solve
61120
.
1
ln
4
ln
4
12
4
)
3
(
5
3
7
1
3
7
1
1








x
x
x
x

In an article concerning predators and prey, Holling
refers to an equation of the form
where x is the prey density, y is the number of prey
attacked, and K and a are constants. Verify his claim
that
Solution:
Find ax first, and thus
Example 5 – Predator-Prey Relation
ax
y
K
K


ln
)
1
( ax
e
K
y 


K
y
K
e
e
K
y
e
K
y
ax
ax
ax









1
)
1
(
ax
y
K
K
ax
K
y
K
ax
K
y
K








ln
ln
ln
(Proved!)

Exponential and Logarithmic Functions Functions

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