Chapter 8
Exponential and
Logarithmic Functions
In this chapter, you will …
• Learn to use exponential functions to
model real-world data.
• Learn to graph exponential functions
and their inverses, logarithmic
functions.
• Learn to solve exponential and
logarithmic equations
8-1 Exploring Exponential Models
• What you’ll learn …
• To model exponential growth
• To model exponential decay
2.03 Use exponential functions to model and solve problems; justify results.
•
Solve using tables, graphs, and algebraic properties.
•
Interpret the constants and coefficients in the context of the problem.
2.04 Create and use best-fit mathematical models of linear, exponential, and
quadratic functions to solve problems involving sets of data.
•
Interpret the constants, coefficients, and bases in the context of the
data.
•
Check the model for goodness-of-fit and use the model, where
appropriate, to draw conclusions or make predictions.
An exponential
function is a function
with the general form
of y = abx where x is a
real number, a ≠ 0,
b > 0, and b ≠ 1.
You can use an
exponential function to
model growth when
b > -1. When b > 1, b is
the growth factor.
Example 1 Graphing Exponential Growth
y = 2x
x
-3
-2
-1
0
1
2
3
y
Real World Connection
b)
Suppose the rate of
increase continues to be
1.24%. Write a function
to model US population
growth.
300
281
250
Population (millions)
Refer to the graph. In 2000,
the annual rate of increase
in the US population was
about 1.24%.
a)
Find the growth factor
for the US population.
200
151
150
100
76
50
23
5
0
1800
1850
1900
Year
1950
2000
Example 3 Writing an Exponential Function
Write an exponential function y = abx for a graph that
includes (2,2) and (3,4).
Write an exponential function y = abx for a graph that
includes (2,4) and (3,16).
An exponential
function can be
used to model
decay, when
0 < b < 1. When
b < 1, b is the
decay factor.
Example 4 Analyzing a Function
Without graphing, determine whether the
function y = 14(0.95)x represents exponential
growth or exponential decay.
Without graphing, determine whether the
function y = 0.2(5)x represents exponential
growth or exponential decay.
An asymptote is a line that a graph
approaches as x or y increases in
absolute value.
Example 5a Graphing Exponential Decay
y = 24(1/3)x
Identify.
x
a. Critical point
-3
-2
-1
b. Horizontal asymptote
c. Domain
0
1
d. range
2
3
y
Example 5b Graphing Exponential Decay
y = 100(0.1)x
Identify.
x
a. Critical point
-3
-2
-1
b. Horizontal asymptote
c. Domain
0
1
d. range
2
3
y
The exponential decay
graph shows the expected
depreciated for a car over
four years. Estimate the
value of the car after 6
years.
The decay factor b = 1 + r,
where r is the annual rate
of decrease. The initial
value of the car is
$20,000. After one year
the value of the car is
about $17,000.
Value ($)
Example 6 Real World Connection
20,000
15,000
10,000
5,000
0 1 2 3 4
Years since purchase
8-2 Properties of Exponential
Functions
• What you’ll learn …
• To identify the role of constants in y=abcx
• To use e as a base
2.03 Use exponential functions to model and solve
problems; justify results.
• Solve using tables, graphs, and algebraic properties.
• Interpret the constants and coefficients in the context
of the problem.
• So far we have graphed functions of
the form y = abx for values of a greater
than 0. When a < 0, the graph of
y = abx is a reflection over the x-axis.
Graph and give asymptote, critical point, domain and range.
y = ½ *2 x
y = (1/3)x
• You can graph many exponential
functions as translations of the parent
function y = abx.
• The graph of y = abx+k + d is the graph
of y = abx translated k units left or
right and d units up or down.
Example 2 Translating y = abx
y =8(1/2)x
y = 8(1/2)x+2 +3
Example 2b Translating y = abx
y =2(3)x-1 + 1
y = -3(4)x+1 +2
Half-life
What does that mean?
• The half-life is the amount of time it
takes for half of the atoms in a sample
to decay.
A = A0(1/2) t/k where k is half-life.
Example 3 Real World Connection
A hospital prepares a 100-mg supply of technetium-99m, which has a half-life of 6
hours. Make a table showing the amount of technetium-99m that remains at the
end of each 6-hour interval for 36 hours. Then write an exponential function to
find the amount of technetium-99m that remains after 75 hours.
Number of 6 hour intervals
0
1
2
3
4
5
6
Number of hours elapsed
0
6
12
18
24
30
36
100
50
25
12.5
6.25
3.13
1.56
Technetuim-99m
Relate The amount of technetium-99m is an exponential function of the number of
half-lives. The initial amount is 100 mg. The decay factor is ½. One half-life
equals 6 hours.
Define Let y = the amount of technetium-99m.
Let x = the number of hours elapsed. Then 1/6x = the number of half-lives.
The Number …. e?
≈ 2.271828
• Exponential functions with a base of e
are useful for describing continuous
growth or decay. Your graphing
calculator has a key for ex.
Graph y = ex
• Find e3
• Find e-3
• Find e1/4
• Find e√x
• In previous courses you have studied
simple interest and compound interest.
Simple Interest
Compound Interest
• The more frequently interest is
compounded, the more quickly the
amount in an account increases. The
formula for continuously compounded
interest uses the number e.
Example 5 Real World Connections
A = Pert
Suppose you invest $1050 at an annual interest rate of 5.5%
compounded continuously. How much will you have in the
account after 5 years?
Suppose you invest $1300 at an annual interest rate of 4.3%
compounded continuously. How much will you have in the
account after three years?
8-3 Logarithmic Functions
as Inverses
• What you’ll learn …
• To write and evaluate logarithmic expressions
• To graph logarithmic functions
1.01 Simplify and perform operations with rational
exponents and logarithms (common and natural) to
solve problems.
2.01 Use the composition and inverse of functions to
model and solve problems; justify results.
• The exponents used by the Richter scale are
called logarithms or logs.
Definition
Logarithm
The logarithm to the base b of a positive
number y is defined as follows:
If y = bx, then logb y = x.
Example 2 Writing in Logarithmic Form
• Write 25 = 52
• Write 729 = 36
• Write (1/2)3 = 1/8
• Write 100 = 1
Example 3 Evaluating Logarithms
• Evaluate log8 16
• Evaluate log9 27
• Evaluate log64 1/32
• Evaluate log10 100
• A common logarithm is a logarithm that uses
base 10. You can write the common
logarithm log10 y as log y.
• Scientist use common logarithms to measure
acidity, which increases as the concentration
of hydrogen ions in a substance increases.
The pH of a substance equals –log[H+], where
[H+] is the concentration of hydrogen ions.
Example 4 Real World Connection
The pH of lemon juice is 2.3, while the pH of
milk is 6.6. Find the concentration of hydrogen
ions in each substance. Which substance is
more acidic?
• A logarithmic function is the inverse of
an exponential function.
Example 5a Graphing a Logarithmic Function
Graph y = log2 x.
Step 1 Graph y=2x.
Step 2 Draw y=x.
Step 3 Choose points y=2x.
Then reverse the coordinates
and plot the points of y = log2 x.
Example 5b Graphing a Logarithmic Function
Graph y = log3 x.
Step 1 Graph y=3x.
Step 2 Draw y=x.
Step 3 Choose points y=3x.
Then reverse the coordinates
and plot the points of y = log3 x.
Translations of Logarithmic Functions
Characteristic
y=logbx
y=logb(x-h) +k
Asymptote
x=0
x-h=0 or x=h
Domain
x>0
x>h
Range
All real
numbers
All real
numbers
Example 6 Translating y=logbx
Graph y=log3 x
Graph y=log6(x-2)+3
8-4 Properties of Logarithms
What you’ll learn …
• To use the properties of logarithms
1.01 Simplify and perform operations with rational
exponents and logarithms (common and natural) to
solve problems.
Simplify
• log2 4 + log2 8
• log3 9 + log3 27
• log2 16 + log2 64
Investigation
1.
Complete the table. Round to the nearest
thousandth.
x
1
2
3
4
5
6
7
8
9
10 15
20
log x
2.
a.
b.
c.
d.
Complete each pair of statements. What do you
notice?
log 3+log 5= ___ and log (3 * 5) = ____
log 1 + log 7 = ___ and log (1 *7) = ____
log 2 + log 4 = ___ and log (2 *4) = ____
log 10 + log 2 = ____ and log (10 *2) = ____
Investigation continued
3.
Complete the statement: log M + log N = __________.
4.
A. Make a conjecture. How could you rewrite the
expression log M using the expressions log M and
N
log N?
B. Use your calculator to verify your conjecture for
several values of M and N.
Properties of Logarithms
For any positive numbers M, N, and b, b ≠1,
logb MN = logb M + logb N
logb
M
N
= logb M - logb N
logb Mx = x logb M
Product Property
Quotient Property
Power Property
Example 1 Identifying the Properties of Logarithms
a. log2 8 – log2 4 = log2 2
b. log5 2 + log5 6 = log5 12
c. 3 logb 4 – 3 logb 2 = logb 8
Example 2 Simplifying Logarithms
a. log3 20 – log3 4
b. 3 log 2 + log 4 – log 16
In Class
• Page 449 1-17 odd
Example 3 Expanding Logarithms
a.
x
log5 y
b. log2 7b
c. log
y
3
2
In Class
• Page 449 19 - 29 odd
• Logarithms are used to
model sound. The intensity
of a sound is a measure of
the energy carried by the
sound wave. The greater the
intensity of a sound, the
louder it seems. This
apparent loudness L is
measured in decibels. You
can use the formula
I
I0
L = 10 log
, where I is the
intensity of the sound in
watts per square meter. I0 is
the lowest intensity sound
that the average human ear
can detect.
8-5 Exponential and Logarithmic
Equations
What you’ll learn …
• To solve exponential equations
• To solve logarithmic equations
1.01 Simplify and perform operations with rational
exponents and logarithms (common and natural) to
solve problems.
2.01 Use the composition and inverse of functions to
model and solve problems; justify results.
Evaluate
• log9 81 * log9 3
• log 10 * log3 9
• log2 16 ÷ log2 8
• Simplify 125-2/3
An equation of the form bcx = a, where
the exponent includes a variable, is an
exponential equation. If m and n are
positive and m=n, then log m = log n.
You can therefore solve an exponential
equation by taking the logarithm of
each side of the equation.
To solve exponential equations
1.
2.
3.
4.
Get the exponent
part by itself.
Take log of both
sides.
Use power property.
Solve for x.
2* 3x + 1 = 9
To solve exponential equations
1.
2.
3.
4.
Get the exponent
part by itself.
Take log of both
sides.
Use power property.
Solve for x.
3x+4 = 101
To solve exponential equations
1.
2.
3.
4.
Get the exponent
part by itself.
Take log of both
sides.
Use power property.
Solve for x.
62x = 21
• To evaluate a logarithm with any base, you
can use the Change of Base Formula.
For any positive numbers M, b, and c, with b ≠ 1 and c ≠ 1,
logb M =
logc M
logc b
Example 2 Using the Change of Base Formula
Use the Change of Base Formula to evaluate log3 15. Then
convert log3 15 to a logarithm in base 2.
Use the Change of Base Formula to evaluate log5 400. Then
convert log5 400 to a logarithm in base 8.
Example 3 Solving an Exponential Equation by
Changing Bases
Solve 23x = 172
Solve 75x = 3000
Example 4 Solving an Exponential Equation by Graphing
Solve 62x = 1500
Solve 116x = 786
Example Real World Connection
The US population of
peninsular bighorn sheep
was 1170 in 1971. By 1999,
only 335 remained.
Write an exponential
equation to model
the decline in the
population. If the
decay rate remains
constant, in what
year might only 5
peninsular bighorn
sheep remain in the
US?
An equation that includes a logarithm
expression is called a logarithmic
equation.
Example 6 Solving a Logarithmic Equation
1. Get log expression by
itself.
2. Rewrite with exponent.
3. Solve for x.
4. Must check (remember
can’t take the log of a
negative)
Solve log (3x+1) = 5
Example 6 Solving a Logarithmic Equation
1. Get log expression by
itself.
2. Rewrite with exponent.
3. Solve for x.
4. Must check (remember
can’t take the log of a
negative)
Solve log (7 – 2x) = -1
Example 6 Solving a Logarithmic Equation
1. Get log expression by
itself.
2. Rewrite with exponent.
3. Solve for x.
4. Must check (remember
can’t take the log of a
negative)
Solve 2 log x- log 3 = 2
Example 6 Solving a Logarithmic Equation
1. Get log expression by
itself.
2. Rewrite with exponent.
3. Solve for x.
4. Must check (remember
can’t take the log of a
negative)
Solve log 6- log 3x = -2
8-6 Natural Logarithms
What you’ll learn …
• To evaluate natural logarithmic expressions
• To solve equations using natural logarithms
1.01 Simplify and perform operations with rational
exponents and logarithms (common and natural) to
solve problems.
2.01 Use the composition and inverse of functions to
model and solve problems; justify results.
In lesson 8.2. we learned that the
number e ≈ 2.71828 can be used as a
base for exponents. The function y = ex
has an inverse, the natural logarithmic
function.
Inverse of f(x) = ln x
x = ln y
y = ex
Find the inverse
1. y = log4 x
2. y = log2 2x
3. y = log (x+1)
4. y = 2log2 x
Example 1 Simplifying Natural Logarithms
1. 3 ln 6 – ln 8
2. 5 ln 2 – ln 4
3. 3 ln x + ln y
4. ¼ ln 3 + ¼ ln x
Example 3 Solving a Natural Logarithm Equation
1. ln (3x + 5) = 4
2. ln x = .1
3. ln (3x – 9) = 21
4. ln
x+2
3
= 12
Example 4 Solving an Exponential Equation
1. 7e2x + 2.5 = 20
2. ex+1 = 30
3. e2x/5 + 7.2 = 9.1
Real World Connection
An initial investment of $100 is now
valued at $149.18. The interest rate is
8% compounded continuously. How
long has the money been invested?
A = Pert
Real World Connection 2
An initial investment of $200 is now
valued at $315.27 after seven years of
continuous compounding. Find the
interest rate.
A = Pert
In this chapter, you should have …
• Learned to use exponential functions
to model real-world data.
• Learned to graph exponential functions
and their inverses, logarithmic
functions.
• Learned to solve exponential and
logarithmic equations