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Algebra 2 Interactive Chalkboard
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Logarithms and Logarithmic Functions
e
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.
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.
b.
c.
Answer: Answer: Answer:
The function represents exponential decay, since the base,
0.5
, is between
0
and
1
.
The function represents exponential growth, since the base,
2
, is greater than
1
.
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 y
with with
10
,
109,478,000
and
a
with
5,283,000
.
Divide each side by
5,283,000
.
Take the
10
th of each side.
root
To find the
10
th root of
20.72
, use selection the MATH menu on the
TI-83
Plus.
Keystrokes:
10 MATH 5 20.72
ENTER under 1.354063324
Answer:
An equation that models the number of cellular telephone subscribers in the U.S. from
1990 2000
is to
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 .
Answer:
Power of a Power Product of Radicals
Simplify each expression.
a.
Answer: b.
Answer:
Solve .
Answer:
The solution is Original equation Rewrite
256
as
4
4 so each side has the same base.
Property of Equality for Exponential Functions Add
2
to each side.
Divide each side by
9
.
Check
Original equation Substitute for
n
.
Simplify.
Simplify.
Solve .
Answer:
The solution is Original equation Rewrite
9
as
3
2 so each side has the same base.
Property of Equality for Exponential Functions Distributive Property Subtract
4
x
each side.
from
Solve each equation.
a.
Answer: b.
Answer:
1
Solve Answer:
The solution is Original inequality Rewrite as Property of Inequality for Exponential Functions Subtract
3
from each side.
Divide each side by
–2
.
Check:
Test a value of
k
less than for example, Original inequality Replace
k
with
0
.
Simplify.
Solve Answer:
Logarithmic to Exponential Form
Exponential to Logarithmic Form
Evaluate Logarithmic Expressions
Inverse Property of Exponents and Logarithms
Solve a Logarithmic Inequality
Solve Equations with Logarithms on Each Side
Solve Inequalities 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 Answer:
So, Let the logarithm equal
y
.
Definition of logarithm Property of Equality for Exponential Functions
Evaluate Answer:
3
Evaluate Answer: .
Evaluate Answer: .
Evaluate each expression.
a.
Answer:
3
b.
Answer:
Solve Answer:
Original equation Definition of logarithm Power of a Power Simplify.
Solve Answer:
9
Solve Check your solution.
Original inequality Logarithmic to exponential inequality Simplify.
Answer:
The solution set is
Check
Try
6
4 to see if it satisfies the inequality.
Original inequality Substitute
6
4 for
x
.
Solve Answer: Check your solution.
Solve
or
Check your solution.
Original equation Property of Equality for Logarithmic Functions Subtract
4
x
add
3
and to each side.
Factor.
Zero Product Property Solve each equation.
Check
Substitute each value into the original equation.
Original equation Substitute
3
for
x
.
Simplify.
Answer:
The solutions are
3
and
1
.
Original equation Substitute
1
for
x
.
Simplify.
Solve Check your solution.
Answer:
The solutions are
3
and
–2
.
Solve
Original inequality Property of Inequality for Logarithmic Functions Addition and Subtraction Properties of Inequalities We must exclude all values of
x
such that Thus the solution set is This compound inequality simplifies to and or
Answer:
The solution set is
Solve Answer:
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 Answer:
Thus,
to approximate the value of
Replace
250
with
5
3 •
2
.
Product Property Inverse Property of Exponents and Logarithms Replace with
0.4307
.
is approximately
3.4307
.
Use Answer:
6.5850
to approximate the value of
Use the value of Answer:
Thus
and to approximate
Replace
4
with the quotient Quotient Property and is approximately
0.7737
.
Use the value of and Answer:
1.2920
to approximate
Sound by The loudness
L
where of a sound in decibels is given
R
is the sound’s relative intensity. The sound made by a lawnmower has a relative intensity of
10 9
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
L
1 be the loudness of one lawnmower running.
Let
L
2 be the loudness of ten lawnmowers running.
Then the increase in loudness is
L
2 –
L
1 .
Substitute for
L
1 and
L
2 .
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 by The loudness
L
where of a sound in decibels is given
R
is the sound’s relative intensity. The sound made by fireworks has a relative intensity of
10 14
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 Answer: approximate the
Replace
216
with
6
3 .
Power Property Replace with
1.1133
.
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
4
th root of each side.
Solve
or
.
Original equation Product Property Definition of logarithm Subtract
64
from each side.
Factor.
Zero Product Property Solve each equation.
Check
Substitute each value into the original equation.
Replace
x
with
–4
.
Since
log
8
(–4)
and
log
8
(–16)
are undefined, extraneous solution and must be eliminated.
–4
is an Replace
x
with
16
.
Product Property
Answer:
The only solution is Definition of logarithm
Solve each equation.
a.
Answer:
12
b.
Answer:
8
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
decimal places.
to four Keystrokes:
LOG 6 ENTER .7781512503
Answer:
about
0.7782
Use a calculator to evaluate log
0.35
decimal places.
to four Keystrokes:
LOG 0.35
ENTER –.4559319557
Answer:
about
–0.4559
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
In
1999
by the equation log an earthquake in Turkey measured
7.4
on the Richter scale. How much energy did this earthquake release? Answer:
about
Solve Answer:
Original equation Property of Equality for Logarithmic Functions Power Property of Logarithms Divide each side by log
62
.
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
Thus,
2.5643
is a reasonable solution. and
3
.
Solve Answer:
2.5789
Solve
Original inequality Property of Inequality for Logarithmic Functions Power Property of Logarithms Distributive Property Subtract 5
x
log from each side.
3
Distributive Property Divide each side by Switch > to < because is negative.
Use a calculator.
Simplify.
Check: Answer:
The solution set is Original inequality Replace
x
with
0
.
Simplify.
Negative Exponent Property
Solve Answer:
Express in terms of common logarithms. Then approximate its value to four decimal places.
Change of Base Formula
Answer:
The value of Use a calculator.
is approximately
2.6309
.
Express in terms of common logarithms. Then approximate its value to four decimal places.
Answer:
Example 1 Evaluate Natural Base Expressions
Example 2 Evaluate Natural Logarithmic Expressions
Example 3 Write Equivalent Expressions
Example 4 Inverse Property of Base
e
e
e
Example 7 Solve Natural Log Equations and Inequalities
Use a calculator to evaluate to four decimal places.
Keystrokes:
2nd [
e x
] 0.5
ENTER 1.648721271
Answer:
about
1.6487
Use a calculator to evaluate to four decimal places.
Keystrokes:
2nd [
e x
] –8 ENTER .0003354626
Answer:
about
0.0003
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 ENTER 1.098612289
Answer:
about
1.0986
Use a calculator to evaluate In to four decimal places.
Keystrokes:
LN 1 ÷ 4 ENTER –1.386294361
Answer:
about
–1.3863
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:
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
r P
with
700
, 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
700
e
(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
reach
$2000
.
years for the balance to
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:
Example 1 Exponential Decay of the Form
y = a
– r
Example 2 Exponential Decay of the Form
y = ae
Example 3 Exponential Growth of the Form
y = a
+ r
Example 4 Exponential Growth of the Form
y = ae
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 Plan
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.
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
130
, and
r
with
13
,
a
with
0.11
.
with 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 of the caffeine to be eliminated from a person’s body.
90%
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
and with
t
with
20
.
130
,
r
with
0.11
Ten percent of
130
seems reasonable.
is
13
, so the answer
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
0.5
a
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
0.1
a
.
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 a
with
2,000,000
, 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
Answer:
D Divide each side by ln
1.03
.
Use a calculator.
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
Answer:
C
D
18
years
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
)
and
U
(
t
)
with with
Property of Inequality for Logarithms Product Property of Logarithms Inverse Property of Exponents and Logarithms Subtract ln 278 0.026
t
and from each side.
Divide each side by –0.017
.
Answer:
Use a calculator.
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
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