INTRODUCTORY MATHEMATICAL ANALYSIS

1. INTRODUCTORY MATHEMATICALINTRODUCTORY MATHEMATICAL
ANALYSISANALYSISFor Business, Economics, and the Life and Social Sciences
©2007 Pearson Education Asia
Chapter 4Chapter 4
Exponential and Logarithmic FunctionsExponential and Logarithmic Functions

2. ©2007 Pearson Education Asia
INTRODUCTORY MATHEMATICAL
ANALYSIS
0. Review of Algebra
1. Applications and More Algebra
2. Functions and Graphs
3. Lines, Parabolas, and Systems
4. Exponential and Logarithmic Functions
5. Mathematics of Finance
6. Matrix Algebra
7. Linear Programming
8. Introduction to Probability and Statistics

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9. Additional Topics in Probability
10. Limits and Continuity
11. Differentiation
12. Additional Differentiation Topics
13. Curve Sketching
14. Integration
15. Methods and Applications of Integration
16. Continuous Random Variables
17. Multivariable Calculus
INTRODUCTORY MATHEMATICAL
ANALYSIS

4. ©2007 Pearson Education Asia
• 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 ObjectivesChapter Objectives

5. ©2007 Pearson Education Asia
Exponential Functions
Logarithmic Functions
Properties of Logarithms
Logarithmic and Exponential Equations
4.1)
4.2)
4.3)
4.4)
Chapter 4: Exponential and Logarithmic Functions
Chapter OutlineChapter Outline

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• 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
.
Chapter 4: Exponential and Logarithmic Functions
4.1 Exponential Functions4.1 Exponential Functions
( ) x
bxf =

7. ©2007 Pearson Education Asia
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,
Chapter 4: Exponential and Logarithmic Functions
4.1 Exponential Functions
Example 1 – Bacteria Growth
( )
t
tN 





=
3
4
200
0
4
(0) 300 300(1) 300
3
N
 
= = = ÷
 
3
4 64 6400
(3) 300 300 711
3 27 9
N
   
= = = ≈ ÷  ÷
   

8. ©2007 Pearson Education Asia
Graph the exponential function f(x) = (1/2)x
.
Solution:
Chapter 4: Exponential and Logarithmic Functions
4.1 Exponential Functions
Example 3 – Graphing Exponential Functions with 0 < b < 1

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Properties of Exponential Functions
Chapter 4: Exponential and Logarithmic Functions
4.1 Exponential Functions

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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
.
Chapter 4: Exponential and Logarithmic Functions
4.1 Exponential Functions
Example 5 – Graph of a Function with a Constant Base
2
Graph 3 .x
y =
(1 )n
S P r= +

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Chapter 4: Exponential and Logarithmic Functions
4.1 Exponential Functions
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) 10,612P = ≈

12. ©2007 Pearson Education Asia
Chapter 4: Exponential and Logarithmic Functions
4.1 Exponential Functions
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
100,000 100,000 100,000 271,828P e e e= = = ≈
0.05
100,000 t
P e=

13. ©2007 Pearson Education Asia
Chapter 4: Exponential and Logarithmic Functions
4.1 Exponential Functions
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−
=
( )
mg100100 0062.0
== −
eN
( )
mg8.53100 10062.0
== −
eN

14. ©2007 Pearson Education Asia
Chapter 4: Exponential and Logarithmic Functions
4.2 Logarithmic Functions4.2 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
=
= =

15. ©2007 Pearson Education Asia
Chapter 4: Exponential and Logarithmic Functions
4.2 Logarithmic Functions
Example 3 – Graph of a Logarithmic Function with b > 1
Sketch the graph of y = log2x.
Solution:

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Chapter 4: Exponential and Logarithmic Functions
4.2 Logarithmic Functions
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.
( ) 210log100log
2
==
01ln =
110log1.0log 1
−== −
1ln1ln 1
−=−=−
ee
2
1
6log2
6log
6log36 ==

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Chapter 4: Exponential and Logarithmic Functions
4.2 Logarithmic Functions
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:
λ
2ln
=T
days
λ
T 4.138
00501.0
2ln2ln
≈==

18. ©2007 Pearson Education Asia
Chapter 4: Exponential and Logarithmic Functions
4.3 Properties of Logarithms4.3 Properties of Logarithms
Example 1 – Finding Logarithms
• Properties of logarithms are:
nmmn bbb loglog)(log.1 +=
nm
n
m
bb logloglog.2 b −=
mrm b
r
b loglog3. =
a.
b.
c.
d.
7482.18451.09031.07log8log)78log(56log =+≈+=⋅=
6532.03010.09542.02log9log
2
9
log =−≈−=
8062.1)9031.0(28log28log64log 2
=≈==
3495.0)6990.0(
2
1
5log
2
1
5log5log 2/1
=≈==
b
m
m
b
m
m
a
a
b
b
b
bb
log
log
log.7
1log.6
01log.5
log
1
log4.
=
=
=
−=

19. ©2007 Pearson Education Asia
Chapter 4: Exponential and Logarithmic Functions
4.3 Properties of Logarithms
Example 3 – Writing Logarithms in Terms of
Simpler Logarithms
a.
b.
wzx
wzx
zwx
zw
x
lnlnln
)ln(lnln
)ln(lnln
−−=
+−=
−=
)]3ln()2ln(8ln5[
3
1
)]3ln()2ln([ln
3
1
)}3ln(])2({ln[
3
1
3
)2(
ln
3
1
3
)2(
ln
3
)2(
ln
85
85
85
3/1
85
3
85
−−−+=
−−−+=
−−−=
−
−
=





−
−
=
−
−
xxx
xxx
xxx
x
xx
x
xx
x
xx

20. ©2007 Pearson Education Asia
Chapter 4: Exponential and Logarithmic Functions
4.3 Properties of Logarithms
Example 5 – Simplifying Logarithmic Expressions
a.
b.
c.
d.
e.
.3ln 3
xe x
=
3
30
10log01000log1log 3
=
+=
+=+
9
89/8
7
9 8
7 7log7log ==
1)3(log
3
3
log
81
27
log 1
34
3
33 −==





=




 −
0)1(1
10logln
10
1
logln 1
=−+=
+=+ −
ee

21. ©2007 Pearson Education Asia
Chapter 4: Exponential and Logarithmic Functions
4.3 Properties of Logarithms
Example 7 – Evaluating a Logarithm Base 5
Find log52.
Solution:
4307.0
5log
2log
2log5log
2log5log
25
≈=
=
=
=
x
x
x
x

22. ©2007 Pearson Education Asia
Chapter 4: Exponential and Logarithmic Functions
4.3 Properties of Logarithms
4.4 Logarithmic and Exponential Equations4.4 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.

23. ©2007 Pearson Education Asia
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.
Chapter 4: Exponential and Logarithmic Functions
4.4 Logarithmic and Exponential Equations
Example 1 – Oxygen Composition
xy log885.0934.5loglog +=

24. ©2007 Pearson Education Asia
Solution:
Chapter 4: Exponential and Logarithmic Functions
4.4 Logarithmic and Exponential Equations
Example 1 – Oxygen Composition
)934.5log(log
log934.5log
log885.0934.5loglog
885.0
885.0
xy
x
xy
=
+=
+=
885.0
934.5 xy =

25. ©2007 Pearson Education Asia
Chapter 4: Exponential and Logarithmic Functions
4.4 Logarithmic and Exponential Equations
Example 3 – Using Logarithms to Solve an
Exponential Equation
Solution:
.124)3(5 1
=+ −x
Solve
61120.1
ln4ln
4
124)3(5
3
71
3
71
1
≈
=
=
=+
−
−
−
x
x
x
x

26. ©2007 Pearson Education Asia
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
Chapter 4: Exponential and Logarithmic Functions
4.4 Logarithmic and Exponential Equations
Example 5 – Predator-Prey Relation
ax
yK
K
=
−
ln
)1( ax
eKy −
−=
K
yK
e
e
K
y
eKy
ax
ax
ax
−
=
−=
−=
−
−
−
1
)1(
ax
yK
K
ax
K
yK
ax
K
yK
=
−
=
−
−
−=
−
ln
ln
ln
(Proved!)