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Homework, Page 296
Tell whether the function is an exponential growth or an
exponential decay function and find the constant percentage
rate of growth or decay.
1. P  t   3.5 1.09t
P  t   3.5 1.09t  1.09  1.00  0.09 100%  9%
P  t  is an exponential growth function.
The constant percentage rate of growth is 9%.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 3- 1
Homework, Page 296
Tell whether the function is an exponential growth or an
exponential decay function and find the constant percentage
rate of growth or decay.
5.
g  t   247 2t
g  t   247 2t  2.00  1.00  1.00  100%  100%
g  t  is an exponential growth function.
The constant percentage rate of growth is 100%.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 3- 2
Homework, Page 296
Determine the exponential function that satisfies the given
conditions.
9. Initial value = 16, decreasing at a rate of 50% per month
f  t   a bt  f  0   a b 0  16  a  16
50%
b  1
 0.5  f  t   16 0.5t
100%
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Slide 3- 3
Homework, Page 296
Determine the exponential function that satisfies the given
conditions.
13. Initial height = 16cm, growing at a rate of 5.2% per week
h  t   a bt  h  0   a b 0  18  a  18
5.2%
1
 1.052  h  t   18 1.052t
100%
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Slide 3- 4
Homework, Page 296
Determine the exponential function that satisfies the given
conditions.
17. Initial mass = 592 g, halving once every six years
m  t   a bt  m  0   a b 0  592  a  592
1
b  
 2
t
t
6
1
 m  t   592  
 2
t
6
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Slide 3- 5
Homework, Page 296
Determine a formula for the exponential function whose graph is
given.
y
21.
(5, 8.05)
(0, 4)
x
f  x   a b x  f  0   a b0  4  a  4  f  5   4 b5  8.05
1 8.05
ln 4  ln b  ln 8.05  5ln b  ln 8.05  ln 4  ln b  ln
5
4
5
be
1 8.05
ln
5
4
 1.150  f  x   4 1.150 x
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 3- 6
Homework, Page 296
Find the logistic function that satisfies the given conditions.
25. Initial population = 16, maximum sustainable
population = 128, passing through  5, 32 
128
128
4
P t  
 32 
1
kt
k5
1  7e
1  7e
1  7e k 5
3
3
k5
k5
k5
k5
1  7e  4  7e  3  e   ln e  ln
7
7
3
3
1 3
5k ln e  ln  5k  ln  k  ln  k  0.169
7
7
5 7
128
P t  
1  7e 0.169t
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 3- 7
Homework, Page 296
Find the logistic function that satisfies the given conditions.
25. Initial population = 16, maximum sustainable
population = 128, passing through  5, 32 
128
128
P t  
 32 
kt
1  7e
1  7e k 5
128
P t  
1  7e 0.169t
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 3- 8
Homework, Page 296
29. The 2000 population of Jacksonville, FL was 736,000
and was increasing at the rate of 1.49% each year. At that
rate, when will the population be one million?
P  t   736000 1.0149t where t = 0 represents 2000
1000000  736000 1.0149t  ln1000000  ln 736000  ln1.0149t
1
1000000
ln1000000  ln 736000  t ln1.0149  t 
ln
ln1.0149
736000
1
1000
t
ln
 20.725  Late 2020
ln1.0149
736
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 3- 9
Homework, Page 296
29. The 2000 population of Jacksonville, FL was 736,000
and was increasing at the rate of 1.49% each year. At that
rate, when will the population be one million?
P  t   736000 1.0149t where t = 0 represents 2000
1000000  736000 1.0149t  Late 2020
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 3- 10
Homework, Page 296
33. The half-life of a radioactive substance is 14 days.
there are 6.6 g initially.
a. Express the amount of the substance remaining as a function
t
of time t.
1
A  t   6.6  
2
14
b. When will there be less than 1 g remaining?
After 39 days.
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Slide 3- 11
Homework, Page 296
37. Using the population model that is graphed, explain
why the time it takes the population to double is
independent of the population size.
The formula for the population is in the form P  t   a 2 c ,
t
where a is the initial population, and c is the number of
years it takes for the population to double.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 3- 12
Homework, Page 296
41. Determine the atmospheric pressure outside an
aircraft flying at 52,800 ft (10 miles above sea level).
1
P  h   14.7  
 2
h
3.6
10
1
P 10   14.7  
 2
3.6
 P 10   2.143 psi
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 3- 13
Homework, Page 296
45. The number of students infected with flu at
Springfield High after t days is modeled by the function:
800
P t  
1  49e0.2t
A. What was the initial number of infected students?
800
800
800
P  0 


 16
0.2 0
1  49 1 50
1  49e
B. When will the number of infected students be 200?
C. The school will close when 300 of the 800-student
body are infected. When will the school close?
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 3- 14
Homework, Page 296
45. B.
When will the number of infected
students be 200?
800
200 
 t  13.966  14 days
0.2 t
1  49e
C. The school will close when 300 of the 800student body are infected. When will the
school close?
800
300 
 t  16.9  day 17
0.2 t
1  49e
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 3- 15
Homework, Page 296
53. What is the constant percentage growth rate of
P  t   1.23 1.049t
a.
b.
c.
d.
e.
49%
23%
4.9%
2.3%
1.23%
1.049  1.000  0.049 100%  4.9%
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 3- 16
Homework, Page 296
57. a. Use the data in the table and logistic regression to
predict the population in 2000.
The logistic model predicts a population of 281.1 million
people in the year 2000.
b. Compare the prediction with the value listed in the
table for 2000.
The model underestimates the population by 0.3 million
people.
c. Which model, logistic or exponential makes the better
prediction in this case?
The logistic model makes a much more accurate estimate
than the exponential model (overestimates by 3 million).
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 3- 17
3.3
Logarithmic Functions and Their
Graphs
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Quick Review
Evaluate the expression without using a calculator.
1. 6
8
2.
2
3. 7
Rewrite as a base raised to a rational number exponent.
1
4.
e
-2
11
32
0
3
5. 10
4
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 3- 19
Quick Review Solutions
Evaluate the expression without using a calculator.
1
1. 6
36
8
2.
2
2
3. 7 1
Rewrite as a base raised to a rational number exponent.
1
4.
e
e
-2
11
32
0
3 / 2
3
5. 10
4
1/ 4
10
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 3- 20
What you’ll learn about





Inverses of Exponential Functions
Common Logarithms – Base 10
Natural Logarithms – Base e
Graphs of Logarithmic Functions
Measuring Sound Using Decibels
… and why
Logarithmic functions are used in many applications, including
the measurement of the relative intensity of sounds.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 3- 21
Changing Between Logarithmic and
Exponential Form
If x  0 and 0  b  1, then y  logb ( x) if and only if b y  x.
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Slide 3- 22
Inverses of Exponential Functions
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 3- 23
Basic Properties of Logarithms
For 0  b  1, x  0, and any real number y.
log b 1  0
because b  1.
0
log b b  1 because b  b.
1
log b b  y
y
b
log b x
x
because b  b .
y
y
because log b x  log b x.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 3- 24
Example Evaluating Logarithms
Evaluate the logarithmic expression without using a calculator.
1
log 6 5
36
10log 25 
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Slide 3- 25
An Exponential Function and Its Inverse
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Slide 3- 26
Common Logarithm – Base 10



Logarithms with base 10 are called common
logarithms.
The common logarithm log10x = log x.
The common logarithm is the inverse of the
exponential function y = 10x.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 3- 27
Basic Properties of Common Logarithms
Let x and y be real numbers with x  0.
log1  0 because 10  1.
0
log10  1 because 10  10.
1
log10  y because 10  10 .
y
10
log x
y
y
 x because log x  log x.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 3- 28
Example Solving Simple Logarithmic
Equations
Solve the equation by changing it to exponential form.
log x  4
log x  5
2
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 3- 29
Basic Properties of Natural Logarithms
Let x and y be real numbers with x  0.
ln1  0 because e  1.
0
ln e  1 because e  e.
1
ln e  y because e  e .
y
e
ln x
y
y
 x because ln x  ln x.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 3- 30
Example Evaluating Natural Logarithms
Evaluate the logarithmic expressions:
ln 23.5 
ln 0.48 
ln  5  
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 3- 31
Graphs of the Common and Natural
Logarithm
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Slide 3- 32
Example Drawing Logarithmic Graphs
Draw the graph of the given function:
ln x
ln   x 
 ln x
 ln   x 
y
x
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 3- 33
Example Transforming Logarithmic
Graphs
Describe how to transform the graph of y  ln x into
the graph of h( x)  ln(2  x).
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 3- 34
Decibels
The level of sound intensity in decibels (dB) is
 I 
  10log   , where  (beta) is the number of decibels,
 I0 
I is the sound intensity in W/m 2 , and I 0  10 12 W/m 2 is the
threshold of human hearing (the quietest audible sound
intensity).
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 3- 35
Example Computing Decibel Levels
Compute the decibel levels of the following
Subway train
Threshold of pain
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Slide 3- 36
Homework


Review Section 3.3
Page 308, Exercises: 1 – 65 (EOO), 59
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Slide 3- 37
3.4
Properties of Logarithmic Functions
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Quick Review
Evaluate the expression without using a calculator.
1. log10
2. ln e
3. log 10
Simplify the expression.
xy
4.
x y
3
3
-2
3
3
2
2
x y
2
5.
4
2x

1/ 2
3
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 3- 39
Quick Review Solutions
Evaluate the expression without using a calculator.
1. log10
3
2. ln e
3
3. log 10
-2
Simplify the expression.
xy
x
4.
x y
y
3
3
-2
3
3
5
2
2
5
x y 
2
5.
4
2x
3
1/ 2
4
x y
2
2
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 3- 40
What you’ll learn about




Properties of Logarithms
Change of Base
Graphs of Logarithmic Functions with Base b
Re-expressing Data
… and why
The applications of logarithms are based on their many
special properties, so learn them well.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 3- 41
Properties of Logarithms
Let b, R, and S be positve real numbers with b  1, and
c any real number.
Product rule:
log b ( RS )  log b R  log b S
Quotient rule:
Power rule:
R
log b    log b R  log b S
S
log b ( R)c  c log b R
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 3- 42
Example Proving the Product Rule for
Logarithms
Prove logb ( RS )  logb R  logb S.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 3- 43
Example Expanding the Logarithm of a
Product
Assuming x is positive, use properties of logarithms to write
 
log 3 x5 as a sum of logarithms or multiple logarithms.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 3- 44
Example Expanding the Logarithm of a
Quotient
Assuming x is positive, use properties of logarithms to write
 3x5 
log 
as a sum or difference of logarithms or

3
 x 5 
multiples of logarithms.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 3- 45
Example Condensing a Logarithmic
Expression
Assuming x is positive, write 3ln x  ln 2  3ln x as a
single logarithm.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 3- 46
Change-of-Base Formula for Logarithms
For positive real numbers a, b, and x with a  1 and b  1,
log a x
log b x 
.
log a b
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 3- 47
Example Evaluating Logarithms by
Changing the Base
Evaluate log3 10.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 3- 48
Example Graphing Logarithmic
Functions
Graph log3 x.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 3- 49
Re-Expression of Data
If we apply a function to one or both of the variables in
a data set, we transform it into a more useful form, e.g.,
in an earlier section we let the numbers 0 – 100
represent the years 1900 – 2000. Such a transformation
is called a re-expression.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 3- 50
Example Re-Expressing Kepler’s Third
Law
Re-express the (a, T) data points in Table 3.20 as (ln a, ln T)
pairs. Find a linear regression model for the re-expressed pairs.
Rewrite the linear regression in terms of a and T, without
logarithms or fractional exponents.
Planet
Mercury
Avg Dist (AU) Period (years)
0.3870
0.2410
Venus
Earth
Mars
0.7233
1.0000
1.523
0.6161
0.0000
1.981
Jupiter
Saturn
5.203
9.539
11.86
29.46
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 3- 51