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Topic 12
Lesson 12-1
Exponential Functions
Lesson 12-2
Logarithms and Logarithmic Functions
Lesson 12-3
Properties of Logarithms
Lesson 12-4
Common Logarithms
Lesson 12-5
Base e and Natural Logarithms
Lesson 12-6
Exponential Growth and Decay
Example 1 Graph an Exponential Function
Example 2 Identify Exponential Growth and Decay
Example 3 Write an Exponential Function
Example 4 Simplify Expressions with
Irrational Exponents
Example 5 Solve Exponential Equations
Example 6 Solve Exponential Inequalities
Sketch the graph of
. Then state the
function’s domain and range.
Make a table of values. Connect the points to sketch a
smooth curve.
x
–2
–1
0
1
2
1
4
16
Answer:
The domain is all real numbers, while
the range is all positive numbers.
Sketch the graph of
Then state the
function’s domain and range.
Answer:
The domain is all real numbers;
the range is all positive numbers.
Using technology
Determine whether
growth or decay.
represents exponential
Answer: The function represents exponential decay,
since the base, 0.7, is between 0 and 1.
Determine whether
growth or decay.
represents exponential
Answer: The function represents exponential growth,
since the base, 3, is greater than 1.
Determine whether
growth or decay.
represents exponential
Answer: The function represents exponential growth,
since the base,
is greater than 1.
Determine whether each function represents
exponential growth or decay.
a.
Answer: The function represents
exponential decay, since the
base, 0.5, is between 0 and 1.
b.
Answer: The function represents
exponential growth, since
the base, 2, is greater than 1.
c.
Answer: The function represents
exponential decay, since the
base,
is between 0 and 1.
Cellular Phones In December of 1990, there were
5,283,000 cellular telephone subscribers in the United
States. By December of 2000, this number had risen to
109,478,000.
Write an exponential function of the form
that
could be used to model the number of cellular
telephone subscribers y in the U.S. Write the function
in terms of x, the number of years since 1990.
For 1990, the time x equals 0, and the initial number of
cellular telephone subscribers y is 5,283,000. Thus the
y-intercept, and the value of a, is 5,283,000.
For 2000, the time x equals 2000 – 1990 or 10,
and the number of cellular telephone subscribers
is 109,478,000.
Substitute these values and the value of a into an
exponential function to approximate the value of b.
Exponential function
Replace x with 10,
y with 109,478,000
and a with 5,283,000.
Divide each side
by 5,283,000.
Take the 10th root
of each side.
To find the 10th root of 20.72, use selection
the MATH menu on the TI-83/84 Plus.
Keystrokes: 10 MATH 5
under
20.72 ENTER 1.354063324
Answer: An equation that models the number of cellular
telephone subscribers in the U.S. from 1990 to
2000 is
Suppose the number of telephone subscribers
continues to increase at the same rate. Estimate the
number of US subscribers in 2010.
For 2010, the time x equals 2010 – 1990 or 20.
Modeling equation
Replace x with 20.
Use a calculator.
Answer: The number of cell phone subscribers will be
about 2,136,000,000 in 2010.
Health In 1991, 4.9% of Americans had diabetes. By
2000, this percent had risen to 7.3%.
a. Write an exponential function of the form
could be used to model the percentage of Americans
with diabetes. Write the function in terms of x, the
number of years since 1991.
Answer:
b. Suppose the percent of Americans with diabetes
continues to increase at the same rate. Estimate the
percent of Americans with diabetes in 2010.
Answer: 11.4%
Simplify
Answer:
.
Quotient of Powers
Simplify
.
Power of a Power
Answer:
Product of Radicals
Simplify each expression.
a.
Answer:
b.
Answer:
Solve
.
Original equation
Rewrite 256 as 44 so each
side has the same base.
Property of Equality for
Exponential Functions
Add 2 to each side.
Divide each side by 9.
Answer: The solution is
Check
Original equation
Substitute
Simplify.
Simplify.
for n.
Solve
.
Original equation
Rewrite 9 as 32 so each
side has the same base.
Property of Equality for
Exponential Functions
Distributive Property
Subtract 4x from
each side.
Answer: The solution is
Solve each equation.
a.
Answer:
b.
Answer: 1
Solve
Original inequality
Rewrite
as
Property of Inequality for
Exponential Functions
Subtract 3 from each side.
Divide each side by –2.
Answer: The solution is
Check: Test a value of k less than
for example,
Original inequality
Replace k with 0.
Simplify.
Solve
Answer:
Topic 12 - 1
Example 1 Logarithmic to Exponential Form
Example 2 Exponential to Logarithmic Form
Example 3 Evaluate Logarithmic Expressions
Example 4 Inverse Property of Exponents
and Logarithms
Example 5 Solve a Logarithmic Equation
Example 6 Solve a Logarithmic Inequality (SKIP)
Example 7 Solve Equations with Logarithms on
Each Side
Write
Answer:
in exponential form.
Write
Answer:
in exponential form.
Write each equation in exponential form.
a.
Answer:
b.
Answer:
Write
Answer:
in logarithmic form.
Write
Answer:
in logarithmic form.
Write each equation in logarithmic form.
a.
Answer:
b.
Answer:
Evaluate
Let the logarithm equal y.
Definition of logarithm
Property of Equality for
Exponential Functions
Answer: So,
Evaluate
Answer: 3
Evaluate
Answer:
.
Evaluate
Answer:
.
Evaluate each expression.
a.
Answer: 3
b.
Answer:
Solve
Original equation
Definition of logarithm
Power of a Power
Answer:
Simplify.
Solve
Answer: 9
Solve
Check your solution.
Original inequality
Logarithmic to
exponential inequality
Simplify.
Answer: The solution set is
Check
Try 64 to see if it satisfies the inequality.
Original inequality
Substitute 64 for x.
Solve
Answer:
Check your solution.
Solve
Check your solution.
Original equation
Property of Equality for
Logarithmic Functions
Subtract 4x and
add 3 to each side.
Factor.
or
Zero Product Property
Solve each equation.
Check Substitute each value into the original equation.
Original equation
Substitute 3 for x.
Simplify.
Original equation
Substitute 1 for x.
Simplify.
Answer: The solutions are 3 and 1.
Solve
Check your solution.
Answer: The solutions are 3 and –2.
Topic 12 - 2
Example 1 Use the Product Property
Example 2 Use the Quotient Property
Example 3 Use Properties of Logarithms
Example 4 Power Property of Logarithms
Example 5 Solve Equations Using Properties
of Logarithms
Use
to approximate the value of
Replace 250 with 53 • 2.
Product Property
Inverse Property
of Exponents
and Logarithms
Replace
with 0.4307.
Answer: Thus,
is approximately 3.4307.
First Property of Logarithms
logb (mn)  logb m  logb n
log 2 (35)  log 2 (7  5)  log 2 7  log 2 5
logb m  logb n  logb (mn)
log 2 7  log 2 5  log 2 (7  5)  log 2 (35)
Answer: Thus,
is approximately 3.4307.
Use
Answer: 6.5850
to approximate the value of
Use
the value of
and
to approximate
Replace 4 with the
quotient
Quotient Property
and
Answer: Thus
is approximately 0.7737.
Second Property of Logarithms
m
log b    logb m  logb n
n
 14 
log 2    log 2 14  log 2 5
 5
m
log b m  log b n  logb  
n
 36 
log 3  36   log 3  4   log 3  
 4 
Use
the value of
Answer: 1.2920
and
to approximate
Sound The loudness L of a sound in decibels is given
by
where R is the sound’s relative
intensity. The sound made by a lawnmower has a
relative intensity of 109 or 90 decibels. Would the
sound of ten lawnmowers running at that same
intensity be ten times as loud or 900 decibels?
Explain your reasoning.
Let L1 be the loudness of one lawnmower running.
Let L2 be the loudness of ten lawnmowers running.
Then the increase in loudness is L2 – L1.
Substitute for L1 and L2.
Product Property
Distributive Property
Subtract.
Inverse Property of
Exponents and Logarithms
Answer: No; the sound of ten lawnmowers is perceived
to be only 10 decibels as loud as the sound of
one lawnmower, or 100 decibels.
Sound The loudness L of a sound in decibels is given
by
where R is the sound’s relative
intensity. The sound made by fireworks has a relative
intensity of 1014 or 140 decibels. Would the sound of
ten fireworks of that same intensity be ten times as
loud or 1400 decibels? Explain your reasoning.
Then the increase in loudness is L2 – L1.
L2  L1  10 Log10 10  1014   10 Log101014
Substitute for L1 and L2
 10  Log1010  Log1010
14
  10Log
10
14
10
Product Property
 10 Log1010  10 Log1010  10 Log1010
14
14
Distributive Property
 10 Log1010  10 Log1010  10 Log1010
14
14
Distributive Property
 10 Log1010
Subtract
Inverse Property of
Exponents and Logarithms
Sound The loudness L of a sound in decibels is given
by
where R is the sound’s relative
intensity. The sound made by fireworks has a relative
intensity of 1014 or 140 decibels. Would the sound of
ten fireworks of that same intensity be ten times as
loud or 1400 decibels? Explain your reasoning.
Answer: No; the sound of ten fireworks is perceived to
be only 10 more decibels as loud as the sound
of one firework, or 150 decibels.
Given that
value of
approximate the
Replace 216 with 63.
Power Property
Answer:
Replace
with 1.1133.
Third Property of Logarithms: Floating Exponent
logb m  n logb m
n
log 2 5  6log 5
6
n log b m  log b m
6log 2 5  log 2 5
6
n
Given that
value of
Answer: 5.1700
approximate the
Solve
.
Original equation
Power Property
Quotient Property
Property of Equality for
Logarithmic Functions
Multiply each side by 5.
Answer:
Take the 4th root
of each side.
Solve
.
Original equation
Product Property
Definition of logarithm
Subtract 64 from each side.
Factor.
or
Zero Product Property
Solve each equation.
Check Substitute each value into the original equation.
Replace x with –4.
Since log8 (–4) and log8 (–16) are undefined, –4 is an
extraneous solution and must be eliminated.
Replace x with 16.
Product Property
Definition of logarithm
Answer: The only solution is
Solve each equation.
a.
Answer: 12
b.
Answer: 8
Topic 12 - 3
Example 1 Find Common Logarithms
Example 2 Solve Logarithmic Equations
Using Exponentiation
Example 3 Solve Exponential Equations
Using Logarithms
Example 4 Solve Exponential Inequalities
Using Logarithms
Example 5 Change of Base Formula
Use a calculator to evaluate log 6 to four
decimal places.
Keystrokes:
LOG
Answer: about 0.7782
6
ENTER
.7781512503
Use a calculator to evaluate log 0.35 to four
decimal places.
Keystrokes:
LOG
0.35 ENTER
Answer: about –0.4559
–.4559319557
Use a calculator to evaluate each expression to four
decimal places.
a. log 5
Answer: 0.6990
b. log 0.62
Answer: –0.2076
Earthquake The amount of energy E, in ergs, that an
earthquake releases is related to its Richter scale
magnitude M by the equation log
The San Fernando Valley earthquake of 1994
measured 6.6 on the Richter scale. How much energy
did this earthquake release?
Write the formula.
Replace M with 6.6.
Simplify.
Write each side
using 10 as a base.
Inverse Property of
Exponents and Logarithms
Use a calculator.
Answer: The amount of energy released was about
ergs.
Earthquake The amount of energy E, in ergs, that
an earthquake releases is related to its Richter scale
magnitude M by the equation log
In 1999 an earthquake in Turkey measured 7.4 on the
Richter scale. How much energy did this earthquake
release?
Answer: about
Solve
Original equation
log 5 5  log 5 62
x
x  log5 62
Take the log5 of both sides
Property of Logarithms
Change of base.
Answer:
Use a calculator.
Check You can check this answer by using a
calculator or by using estimation. Since
and
the value of x is between 2 and 3.
Thus, 2.5643 is a reasonable solution.
Solve
Answer: 2.5789
Solve
Original inequality
Property of Inequality for
Logarithmic Functions
Power Property
of Logarithms
Distributive Property
Subtract 5x log 3
from each side.
x  7log 2  5log3  3log3
Factor an x.
Divide each side by
Switch > to < because
is negative.
Use a calculator.
Simplify.
Check:
Original inequality
Replace x with 0.
Simplify.
Negative Exponent
Property
Answer: The solution set is
Solve
5  10
3x
x 2
log 5  log10
3x
x 2
Original inequality
Property of Inequality for
Logarithmic Functions
3x log 5  ( x  2) log10
Power Property
of Logarithms
3x log 5  x log10  2 log10
3x log 5  x log10  2 log10
Distributive Property
Subtract x log 10
from each side.
Solve
3x log 5  x log10  2 log10
x(3log 5  log10)  2 log10
2 log10
x
(3log 5  log10)
Subtract x log 10
from each side.
Factor an x
Divide both sides by
(3log5 – log10)
DON’T Switch from < to >
Answer:
3log5 – log10 IS NOT negative
Express
in terms of common logarithms. Then
approximate its value to four decimal places.
Change of Base Formula
Use a calculator.
Answer: The value of
is approximately 2.6309.
Express
in terms of common logarithms. Then
approximate its value to four decimal places.
Answer:
Topic 12 - 4
Example 1 Evaluate Natural Base Expressions
Example 2 Evaluate Natural Logarithmic Expressions
Example 3 Write Equivalent Expressions
Example 4 Inverse Property of Base e and Natural
Logarithms
Example 5 Solve Base e Equations
Example 6 Solve Base e Inequalities
Example 7 Solve Natural Log Equations and Inequalities
Use a calculator to evaluate
Keystrokes:
2nd
[ex] 0.5
Answer: about 1.6487
to four decimal places.
ENTER
1.648721271
Use a calculator to evaluate
Keystrokes:
Ti-83/84
2nd
LN
e^ ( –8)
Answer: about 0.0003
to four decimal places.
ENTER
.0003354626
Use a calculator to evaluate each expression to four
decimal places.
a.
Answer: 1.3499
b.
Answer: 0.1353
Use a calculator to evaluate In 3 to four decimal places.
Keystrokes:
LN
3
Answer: about 1.0986
ENTER
1.098612289
Use a calculator to evaluate In
Keystrokes:
LN
1÷4
Answer: about –1.3863
to four decimal places.
ENTER
–1.386294361
Use a calculator to evaluate each expression to four
decimal places.
a. In 2
Answer: 0.6931
b. In
Answer: –0.6931
Write an equivalent logarithmic equation for
Answer:
.
Write an equivalent exponential equation for
Answer:
Write an equivalent exponential or logarithmic equation.
a.
Answer:
b.
Answer:
Evaluate
Answer:
, using your calculator.
Evaluate
Answer:
.
Evaluate each expression.
a.
Answer: 7
b.
Answer:
Solve
Original equation
Subtract 4 from each side.
Divide each side by 3.
Property of Equality
for Logarithms
Inverse Property of
Exponents and Logarithms
Divide each side by –2.
Use a calculator.
Answer: The solution is about –0.3466.
Check You can check this value by substituting –0.3466
into the original equation or by finding the intersection of
the graphs of
and
Solve
Answer: 0.8047
Savings Suppose you deposit $700 into an account
paying 6% annual interest, compounded continuously.
What is the balance after 8 years?
Continuous compounding formula
Replace P with 700,
r with 0.06, and t with 8.
Simplify.
Use a calculator.
Answer: The balance after 8 years would be $1131.25.
How long will it take for the balance in your account to
reach at least $2000?
The balance is at least $2000.
A

2000
Replace A with 700e(0.06)t.
Write an inequality.
Divide each side by 700.
Property of Inequality
for Logarithms
Inverse Property of
Exponents and
Logarithms
Divide each side by 0.06.
Use a calculator.
Answer: It will take at least 17.5 years for the balance to
reach $2000.
Savings Suppose you deposit $700 into an account
paying 6% annual interest, compounded continuously.
a. What is the balance after 7 years?
Answer: $1065.37
b. How long will it take for the balance in your account to
reach at least $2500?
Answer: at least 21.22 years
Solve
Original equation
Write each side using
exponents and base e.
Inverse Property of
Exponents and Logarithms
Divide each side by 3.
Use a calculator.
Answer: The solution is 0.5496. Check this
solution using substitution or graphing.
Solve
Original inequality
Write each side using
exponents and base e.
Inverse Property of
Exponents and Logarithms
Add 3 to each.
Divide each side by 2.
Use a calculator.
Answer: The solution is all numbers less than
7.5912 and greater than 1.5. Check
this solution using substitution.
Solve each equation or inequality.
a.
Answer: about 1.0069
b.
Answer:
Topic 12 - 5
Example 1 Exponential Decay of the Form y = a(1 – r)t
Example 2 Exponential Decay of the Form y = ae–kt
Example 3 Exponential Growth of the
Form y = a(1 + r )t
Example 4 Exponential Growth of the Form y = aekt
Caffeine A cup of coffee contains 130 milligrams of
caffeine. If caffeine is eliminated from the body at a rate
of 11% per hour, how long will it take for 90% of this
caffeine to be eliminated from a person’s body?
Explore
The problem gives the amount of caffeine
consumed and the rate at which the caffeine
is eliminated. It asks you to find the time it will
take for 90% of the caffeine to be eliminated
from a person’s body.
Plan
Use the formula
Let t be the
number of hours since drinking the coffee.
The amount remaining y is 10% of 130 or 13.
Solve
Exponential decay formula
Replace y with 13, a with
130, and r with 0.11.
Divide each side by 130.
Property of Equality
for Logarithms
Power Property
for Logarithms
Divide each side by log 0.89.
Use a calculator.
Answer: It will take approximately 20 hours for 90%
of the caffeine to be eliminated from a
person’s body.
Examine Use the formula to find how much of the
original 130 milligrams of caffeine would
remain after 20 hours.
Exponential decay formula
Replace a with 130, r with 0.11
and t with 20.
Ten percent of 130 is 13, so the answer
seems reasonable.
Caffeine A cup of coffee contains 130 milligrams of
caffeine. If caffeine is eliminated from the body at a rate
of 11% per hour, how long will it take for 80% of this
caffeine to be eliminated from a person’s body?
Answer: 13.8 hours
Geology The half-life of Sodium-22 is 2.6 years.
What is the value of k for Sodium-22?
Exponential decay formula
Replace y with 0.5a and t with 2.6.
Divide each side by a.
Property of Equality for
Logarithmic Functions
Inverse Property of
Exponents and Logarithms
Divide each side by –2.6.
Use a calculator.
Answer: The constant k for Sodium-22 is 0.2666. Thus,
the equation for the decay of Sodium-22 is
where t is given in years.
A geologist examining a meteorite estimates that it
contains only about 10% as much Sodium-22 as it
would have contained when it reached the surface of
the Earth. How long ago did the meteorite reach the
surface of the Earth?
Formula for the
decay of Sodium-22
Replace y with 0.1a.
Divide each side by a.
Property of Equality
for Logarithms
Inverse Property for
Exponents and Logarithms
Divide each side by –0.2666.
Use a calculator.
Answer: It was formed about 9 years ago.
Health The half-life of radioactive iodine used
in medical studies is 8 hours.
a. What is the value of k for radioactive iodine?
Answer:
b. A doctor wants to know when the amount of
radioactive iodine in a patient’s body is 20%
of the original amount. When will this occur?
Answer: about 19 hours later
Multiple-Choice Test Item
The population of a city of one million is increasing at
a rate of 3% per year. If the population continues to
grow at this rate, in how many years will the
population have doubled?
A 4 years
B 5 years
C 20 years
D 23 years
Read the Test Item
You want to know when the population has doubled
or is 2 million. Use the formula
Solve the Test Item
Exponential growth formula
Replace y with 2,000,000,
a with 1,000,000, and r with 0.03.
Divide each side by 1,000,000.
Property of Equality
for Logarithms
Power Property
of Logarithms
Divide each side by ln 1.03.
Use a calculator.
Answer: D
Multiple-Choice Test Item
The population of a city of 10,000 is increasing at a
rate of 5% per year. If the population continues to
grow at this rate, in how many years will the
population have doubled?
A 10 years
B 12 years
C 14 years
D 18 years
Answer: C
Population As of 2000, Nigeria had an estimated
population of 127 million people and the United States
had an estimated population of 278 million people. The
growth of the populations of Nigeria and the United
States can be modeled by
and
, respectively. According to these
models, when will Nigeria’s population be more than
the population of the United States?
You want to find t such that
Replace N(t) with
and U(t) with
Property of Inequality
for Logarithms
Product Property
of Logarithms
Inverse Property of
Exponents and Logarithms
Subtract ln 278 and
0.026t from each side.
Divide each side by –0.017.
Use a calculator.
Answer: After 46 years or in 2046, Nigeria’s population will
be greater than the population of the U.S.
Population As of 2000, Saudi Arabia had an estimated
population of 20.7 million people and the United States
had an estimated population of 278 million people. The
growth of the populations of Saudi Arabia and the
United States can be modeled by
and
, respectively. According to these
models, when will Saudi Arabia’s population be more
than the population of the United States?
Answer: after 109 years or in the year 2109
Topic 12 - 6
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