Transcript Chapter 4: Exponential and Logarithmic Functions

INTRODUCTORY MATHEMATICAL ANALYSIS
For Business, Economics, and the Life and Social Sciences
Chapter 4
Exponential and Logarithmic Functions
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Chapter 4: Exponential and Logarithmic Functions
Chapter Objectives
• 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.
2011 Pearson Education, Inc.
Chapter 4: Exponential and Logarithmic Functions
Chapter Outline
4.1) Exponential Functions
4.2) Logarithmic Functions
4.3) Properties of Logarithms
4.4) Logarithmic and Exponential Equations
2011 Pearson Education, Inc.
Chapter 4: Exponential and Logarithmic Functions
4.1 Exponential Functions
• The function f defined by f x   b x
where b > 0, b  1, and the exponent x is any real
number, is called an exponential function with
base b1.
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Chapter 4: Exponential and Logarithmic Functions
4.1 Exponential Functions
Example 1 – Bacteria Growth
The number of bacteria present in a culture after t
minutes is given by N t   300  4  .
3
a. How many bacteria are present initially?
b. Approximately how many bacteria are present
after 3 minutes?
t
Solution:
0
4
a. When t = 0, N (0)  300    300(1)  300
3
3
4
64
6400
b. When t = 3, N (3)  300    300   
 711
3
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 27 
9
Chapter 4: Exponential and Logarithmic Functions
4.1 Exponential Functions
Example 3 – Graphing Exponential Functions with 0 < b < 1
Graph the exponential function f(x) = (1/2)x.
Solution:
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Chapter 4: Exponential and Logarithmic Functions
4.1 Exponential Functions
Properties of Exponential Functions
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Chapter 4: Exponential and Logarithmic Functions
4.1 Exponential Functions
Example 5 – Graph of a Function with a Constant Base
Graph y  3 .
x2
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
S  P(1  r )n .
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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 P(3)  10,000(1.02)3  10,612.
2011 Pearson Education, Inc.
Chapter 4: Exponential and Logarithmic Functions
4.1 Exponential Functions
Example 9 – Population Growth
The projected population P of a city is given by
P  100,000e0.05t where t is the number of years after
1990. Predict the population for the year 2010.
Solution:
For t = 20,
P  100,000e0.05(20)  100,000e1  100,000e  271,828
2011 Pearson Education, Inc.
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
N  100e0.062t .
a. How many milligrams are initially present?
Solution: For t = 0, N  100e0.062 0  100 mg.
b. How many milligrams are present after 10 days?
Solution: For t = 10, N  100e0.062 10   53.8 mg.
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Chapter 4: Exponential and Logarithmic Functions
4.2 Logarithmic Functions
• y = logbx
if and only
if by=x.
• Fundamental equations are logb b x  x and blogb x  x
Example 1 – Converting from Exponential to Logarithmic Form
Exponential Form
a. Since 52  25
b. Since 34  81
c. Since 100  1
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Logarithmic Form
it follows that
log5 25  2
it follows that
log3 81  4
it follows that
log10 1  0
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.
log100  log10  2
2
b. Find ln 1.
c. Find log 0.1.
ln 1  0
log0.1  log101  1
d. Find ln e-1.
ln e 1  1ln e  1
d. Find log366.
log 6 1
log36 6 

2 log 6 2
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Chapter 4: Exponential and Logarithmic Functions
4.2 Logarithmic Functions
• If a radioactive element has decay constant λ, the
half-life of the element is given by
T
ln 2

Example 7 – Finding Half-Life
A 10-milligram sample of radioactive polonium 210
(which is denoted 210Po) decays according to the
equation. Determine the half-life of 210Po.
Solution:
ln 2
ln 2
T 

 138.4 days
λ
0.00501
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Chapter 4: Exponential and Logarithmic Functions
4.3 Properties of Logarithms
• Properties of logarithms are:
1
  logb m
m
5. logb 1  0
1. logb (mn)  logb m  logb n
m
2. logb
 logb m  logb n
n
4. logb
6. logb b  1
3. logb mr  r logb m
7. logb m 
Example 1 – Finding Logarithms
loga m
loga b
a. log 56  log(8  7)  log 8  log 7  0.9031 0.8451  1.7482
b. log 9  log9  log2  0.9542  0.3010  0.6532
2
c. log 64  log 82  2log 8  2(0.9031)  1.8062
1
1
1/ 2
d. log 5  log5  log5  (0.6990)  0.3495
2
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2
Chapter 4: Exponential and Logarithmic Functions
4.3 Properties of Logarithms
Example 3 – Writing Logarithms in Terms of Simpler
Logarithms
a. ln x  ln x  ln( zw)
zw
 ln x  (ln z  ln w )
 ln x  ln z  ln w
b.
1/ 3
5
8


x
(
x

2
)
x
(
x

2
)
1
x
(
x

2
)
ln 3
 ln 
  ln
x 3
3
x 3
 x 3 
1
 {ln[ x 5 ( x  2)8 ]  ln( x  3)}
3
1
 [ln x 5  ln( x  2)8  ln( x  3)]
3
1
 [5 ln x  8 ln( x  2)  ln( x  3)]
3
5
8
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5
8
Chapter 4: Exponential and Logarithmic Functions
4.3 Properties of Logarithms
Example 5 – Simplifying Logarithmic Expressions
a. ln e3 x  3 x.
b. log1  log1000  0  log103
03
3
9
8
8/9
log
7

log
7

c.
7
7
8
9
 33 
 27 
d. log3    log3  4   log3 (31 )  1
 81 
3 
e. ln e  log
1
 ln e  log101
10
 1  ( 1)  0
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Chapter 4: Exponential and Logarithmic Functions
4.3 Properties of Logarithms
Example 7 – Evaluating a Logarithm Base 5
Find log52.
5x  2
Solution:
log 5 x  log 2
x log 5  log 2
log 2
x
 0.4307
log 5
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Chapter 4: Exponential and Logarithmic Functions
4.3 Properties of Logarithms
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.
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Chapter 4: Exponential and Logarithmic Functions
4.4 Logarithmic and Exponential Equations
Example 1 – Oxygen Composition
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
log y  log 5.934  0.885 log x
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.
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Chapter 4: Exponential and Logarithmic Functions
4.4 Logarithmic and Exponential Equations
Example 1 – Oxygen Composition
Solution:
log y  log 5.934  0.885 log x
 log 5.934  log x 0.885
log y  log( 5.934 x 0.885 )
y  5.934x 0.885
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Chapter 4: Exponential and Logarithmic Functions
4.4 Logarithmic and Exponential Equations
Example 3 – Using Logarithms to Solve an Exponential
Equation
Solve 5  (3)4x 1  12.
Solution:
5  (3)4 x 1  12
4 x 1 
7
3
ln 4 x 1  ln 73
x  1.61120
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Chapter 4: Exponential and Logarithmic Functions
4.4 Logarithmic and Exponential Equations
Example 5 – Predator-Prey Relation
In an article concerning predators and prey, Holling
refers to an equation of the form y  K (1 eax )
where x is the prey density, y is the number of prey
attacked, and K and a are constants. Verify his claim
K
that
ln
K y
 ax
Solution:
Find ax first,y  K (1  e ax ) and thus
y
 1  e ax
K
K y
e ax 
K
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K y
 ax
K
K y
 ln
 ax
K
K
ln
 ax (Proved!)
K y
ln