Exponential and Logarithmic Equations

C H A P T ER 9 Exponential and Logarithmic Equations © 2010 Carnegie Learning, Inc. 9 The rapid spread of an new infectious disease through a hum...
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C H A P T ER

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Exponential and Logarithmic Equations

© 2010 Carnegie Learning, Inc.

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The rapid spread of an new infectious disease through a human population is called a pandemic. In the United States, the H1N1 influenza pandemic infected 47 million people and caused over 9,800 deaths in 2009. You will use exponential functions to predict the number of people who will be diagnosed with swine flu in the future.

9.1

From Exponentials Come Logarithms Properties of Logarithms | p. 339

9.2

Solving More Equations Exponential and Logarithmic Equations | p. 347

9.3

Do You Have Any Interest? Exponential Functions and Data | p. 351

9.4

Depreciation, Decay, and More Exponential and Logarithmic Problem Solving | p. 357

Chapter 9 l Exponential and Logarithmic Equations

337

© 2010 Carnegie Learning, Inc.

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Chapter 9 l Exponential and Logarithmic Equations

9.1

From Exponentials Come Logarithms Properties of Logarithms

Objectives

Key Terms

In this lesson you will:

l common logarithm

l Derive the properties of logarithms.

l natural logarithm

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l Rewrite exponential relationships

using logarithms.

Problem 1 Deriving Logarithmic Properties By definition, logarithms are exponents. So, logarithms have properties that are similar to exponential properties. In this activity, you will derive the properties of logarithms using the properties of exponents. 1. Simplify each exponential expression. a. 50

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b. x0

5a 0 c. ___ b

( )

2. What is the value of any base raised to the zero power? Write an exponential equation to demonstrate this property.

3. Write the property from Question 2 in logarithmic form.

Lesson 9.1 l Properties of Logarithms

339

4. Write a sentence to summarize the logarithmic property corresponding to the logarithmic equation in Question 3.

5. Simplify each exponential expression. a. 161

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b. r1

x 1 c. ___ 3y

( )

6. What is the value of any base raised to the first power? Write an exponential equation to demonstrate this property.

8. Write a sentence to summarize the logarithmic property corresponding to the logarithmic equation in Question 7.

9. Simplify each exponential expression. a. x2  x4

b. s

s

4

c. y 3  y 5

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Chapter 9 l Exponential and Logarithmic Equations

© 2010 Carnegie Learning, Inc.

7. Write the property from Question 6 in logarithmic form.

10. Write a sentence to summarize the exponential property am

a

n

 amn.

11. To derive the logarithmic property corresponding to the exponential property am  an  amn, let x  am and y  an. Then, perform the following steps. a. Substitute x and y into the equation am

a

n

 amn.

9 b. Write the exponential equation from part (a) in logarithmic form.

c. Write the exponential equations x  am and y  an individually in logarithmic form.

d. Substitute the logarithmic expressions for m and n into the equation from part (b).

e. Write a sentence to summarize the logarithmic property corresponding to the exponential property am  an  amn.

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12. Simplify each exponential expression. x6 a. __ x4

w10 b. ___ w

(x  1)6 c. _______3 (x  1)

Lesson 9.1 l Properties of Logarithms

341

am  amn. 13. Write a sentence to summarize the exponential property ___ an

14. To derive the logarithmic property corresponding to the exponential am  amn, let x  am and y  an. Then, perform the property ___ an following steps. am  amn. a. Substitute x and y into the equation ___ an

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b. Write the exponential equation from part (a) in logarithmic form.

c. Write the equations x  am and y  an in logarithmic form.

d. Substitute the logarithmic expressions for m and n into the equation from part (b).

e. Write a sentence to summarize the logarithmic property corresponding am  a mn. to the exponential property ___ an

a. Write a sentence to summarize the logarithmic property logb a x  x logb a.

b. Derive the logarithmic property logb a x  x logb a. Recall that a x is the base a multiplied by itself x times.

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Chapter 9 l Exponential and Logarithmic Equations

© 2010 Carnegie Learning, Inc.

15. Another logarithmic property can be written as logb a x  x logb a.

A common logarithm is a logarithm with base 10. Common logarithms are written as log with no base. A natural logarithm, abbreviated as ln, is a logarithm with base e. Many calculators only evaluate common logarithms and natural logarithms. To evaluate logarithms with other bases, the logarithm must be converted to an expression using only common logarithms or natural logarithms. 16. To convert log517 to a logarithmic expression using only common logarithms or natural logarithms, perform the following steps. a. Set the logarithm equal to a variable.

9 b. Write the logarithmic equation from part (a) in exponential form.

c. Take the common logarithm of both sides of the exponential equation from part (b).

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d. Solve the equation from part (c) for the variable. Use the properties of logarithms as needed.

17. Convert logb a to a logarithmic expression using only common logarithms or natural logarithms.

Lesson 9.1 l Properties of Logarithms

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18. Write each logarithm using only common logarithms or natural logarithms. a. log4 6

b. log8 145

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c. log3 18.96

19. Write each logarithmic expression in expanded form using the properties of logarithms. a. log4 6x5

3y4 b. log7 ___ x3

20. Write each logarithmic expression using a single logarithm. a. log2 10  3 log2 x

b. 4 log 12  4 log 2

c. 3(In 3  In x)  (In x  In 9)

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Chapter 9 l Exponential and Logarithmic Equations

© 2010 Carnegie Learning, Inc.

c. In 3xy3

Problem 2 Summarizing Logarithmic Properties Complete the following table to summarize the properties of logarithms. Exponential Property

Corresponding Logarithmic Property

x0  1

9 x1  x am  an  amn m

a  amn ___ n a

logb ax  Change of Base Formula

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Be prepared to share your methods and solutions.

Lesson 9.1 l Properties of Logarithms

345

© 2010 Carnegie Learning, Inc.

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Chapter 9 l Exponential and Logarithmic Equations

9.2

Solving More Equations Exponential and Logarithmic Equations

Objectives

Key Term

In this lesson you will:

l logarithmic equation

l Solve logarithmic equations. l Use exponents and logarithms to

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solve equations.

Problem 1 Logarithmic Equations with a Single Logarithm A logarithmic equation is an equation that contains a logarithm. One method to solve a logarithmic equation involving a single logarithm is to rewrite the relationship as an exponential equation and simplify. For example, to solve log2 x  4, perform the following steps. log2 x  4 24  x 16  x Check: log2 16  4

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Solve each logarithmic equation. 1. log3 x  2

2. log5 x  2

3. log2 32  x

4. logx 16  4

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347

5. log x  4.23

6. log 246  x

7. In x  4.23

8. In 246  x

9. log8 145  x

10. logx 24  6.7

Problem 2 Logarithmic Equations with Multiple Logarithms One method to solve a logarithmic equation involving multiple logarithms is to use the properties of logarithms to write each side of the equation as a single logarithm with the same base. Then, use the property that if logx a  logx b, then a  b. For example, to solve log 25  log x  log 5, perform the following steps. log 25  log x  log 5, log 25  log 5x 25  5x 5x Check: log 5  log 5  0.6990  0.6990  1.3979 101.3979  25

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Chapter 9 l Exponential and Logarithmic Equations

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Solve each logarithmic equation. 1. 2 log 6  log x  log 2

2. 2 log(x  1)  log x  log(x  3)

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(

)

4. ln(x  3)  ln(x  2)  ln(2x  24)

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15 3. 2 log(x  3)  log 4  log x  ___ 4

Lesson 9.2 l Exponential and Logarithmic Equations

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5. 2 log3 x  log3 4  log3 16

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© 2010 Carnegie Learning, Inc.

Be prepared to share your methods and solutions.

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Chapter 9 l Exponential and Logarithmic Equations

9.3

Do You Have Any Interest? Exponential Functions and Data

Objectives In this lesson you will: l Model data using exponential functions. l Use logarithms to determine the exponential model of data sets.

9 Problem 1 Interest As We Know It! The principal of a savings account with continuous compound interest is P  P0ert where P is the current principal of the account, P0 is the initial principal of the account, r is the interest rate in decimal form, and t is the time in years. 1. Write a function to represent the principal over time for a continuous compound interest account with an initial principal of $1000 and an interest rate of 2.5%.

© 2010 Carnegie Learning, Inc.

2. Graph the function from Question 1.

Lesson 9.3 l Exponential Functions and Data

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3. Calculate the principal for each time. Then, calculate the natural log of each principal. Time (years)

Principal (dollars)

Natural Log of the Principal

t

P

In P

1 2 5 10

9

20 25

5. Calculate the rate of change between any three pairs of points represented in the scatter plot. What do you notice about the rates of change that you calculated?

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4. Create a scatter plot of the time and the natural log of the principal using the table from Question 3.

6. What type of relationship exists between t and ln P? Explain.

7. Write an equation to represent the relationship between t and ln P.

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8. Rewrite the equation in Question 7 as equal powers of e and simplify the resulting equation.

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9. How can you determine an exponential function to model a data set?

The process to determine an exponential model of a problem situation can use logarithms with any base, but common logarithms and natural logarithms are more often used.

Lesson 9.3 l Exponential Functions and Data

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Problem 2 What’s My Interest? Jackie inherited two continuous compound interest savings accounts from her aunt. The partial statement for the first account is shown.

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Year

Principal at End of Year

28

$13,700.58

29

$14,202.78

30

$14,723.40

Natural Log of Principal at End of Year

She would like to know how much her aunt invested originally and the interest rate. 1. Complete the last column of the table.

2. Calculate the unit rate of change of the natural log of the principal at the end of the year using the first and last row of the table.

4. Rewrite the equation in Question 3 as an exponential equation.

5. How much money did Jackie’s aunt originally invest in the first account? What was the interest rate?

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3. Write an equation to represent the linear relationship between t and ln P.

The partial statement for the second account is shown. Year

Principal at End of Year

5

$3162.27

9

$3816.34

15

$5059.62

Natural Log of Principal at End of Year

6. Complete the last column of the table.

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© 2010 Carnegie Learning, Inc.

7. How much money did Jackie’s aunt originally invest in the second account? What was the interest rate?

Problem 3 Swine Flu Pandemic The number of people who were diagnosed with the swine flu over a 5-week period is displayed in the table.

Week

Number of People Diagnosed with Swine Flu at End of Week

1

1878

3

2579

5

3543

Common Log of Number of People Diagnosed

1. Complete the last column of the table.

Lesson 9.3 l Exponential Functions and Data

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2. Calculate the unit rate of change of the common log of the number of people diagnosed over time using two rows of the table.

3. Write an equation to represent the linear relationship between the number of weeks and the common log of the number of people diagnosed.

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4. Rewrite the equation in Question 3 as equal powers of 10 and simplify the resulting equation.

6. How many people will be diagnosed at the end of week 10?

Be prepared to share your methods and solutions.

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Chapter 9 l Exponential and Logarithmic Equations

© 2010 Carnegie Learning, Inc.

5. How many people were originally diagnosed?

9.4

Depreciation, Decay, and More Exponential and Logarithmic Problem Solving

Objective In this lesson you will: l Determine exponential models for problem situations.

9 Problem 1 Depreciation Depreciation is a decrease, or loss in value. One way to model depreciation is by calculating the percentage decrease in value over time. This model of depreciation can be represented by an exponential function. A 3-year-old luxury car was appraised at $45,250. The same car was appraised at $35,100 when it was 6 years old.

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1. Determine an exponential equation to model the appraisal value of the car over time.

2. How much was the car appraised at when it was new?

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3. How much did the car depreciate in the first year?

4. How much will the car be appraised at when it is 10 years old?

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5. How much will the car be appraised at when it is 20 years old?

© 2010 Carnegie Learning, Inc.

6. How old is the car if it was appraised at $15,000?

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Chapter 9 l Exponential and Logarithmic Equations

Problem 2 Population

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The table displays the population of the United States in millions of people every 10 years starting in 1815. Time Since 1815 (years)

Population (millions of people)

0

8.3

10

11

20

14.7

30

19.7

40

26.7

50

35.2

60

44.4

70

55.9

80

68.9

90

83.2

100

98.8

110

114.2

120

127.1

130

140.1

140

164

150

190.9

160

214.3

Common Log of Population

Unit Rate of Change of Common Log of Population

Population as Predicted by the Exponential Equation

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The population appears to be increasing exponentially. However, real data is rarely a perfect fit. Determine how well an exponential model fits the data by performing the following. 1. Complete the third and fourth columns of the table.

2. What is the average of the unit rates of change?

Lesson 9.4 l Exponential and Logarithmic Problem Solving

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3. Write an exponential equation to model the population over time. Use the average of the unit rates of change and the ordered pair (50, 1.5465).

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4. Complete the last column of the table using the exponential equation derived in Question 3.

6. What could be an explanation for the slowing of the population growth in the 1920s?

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5. How well did the exponential equation from Question 3 model the data? Explain.

Problem 3 Decay The table displays the radioactive decay of an isotope over time.

Time (years)

Amount of Isotope Remaining

0

1000

1

841

2

707

5

420

9

210

10

177

14

88

15

74

Common Log of Amount of Isotope Remaining

Unit Rate of Change of Common Log of Amount of Isotope Remaining

Amount of Isotope Remaining as Predicted by the Exponential Equation

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1. Complete the third and fourth columns of the table.

© 2010 Carnegie Learning, Inc.

2. What is the average of the unit rates of change?

3. Write an exponential equation to model the amount of the isotope remaining over time. Use the average of the unit rates of change and the ordered pair (9, 2.3222).

Lesson 9.4 l Exponential and Logarithmic Problem Solving

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4. Complete the last column of the table using the exponential equation derived from Question 3.

9 5. How well did the exponential equation from Question 3 model the data? Explain.

Be prepared to share your methods and solutions.

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6. The half-life of an isotope is the time that it takes for half of the isotope to remain. What is the half-life of this isotope?