Name.

Class-

Date

Exponential and Logarithmic Equations and Inequalities Essential question: What is the general process for solving exponential and logarithmic equations? An exponential equation is an equation in which the variable appears only as an exponent. The following property is useful for solving some types of exponential equations. Property of Equality for Exponential Equations For any positive number b other than l,ifbx=by, then x = y. CC.9-12.A.SSE.3c

• B E X A M P L E \g Exponential Equations

Algebraically

Solve each exponential equation.

Because the bases are equal, the exponents are equal. Solve for x. _ 27*+!

B

x+ 1

Write both bases as powers with a base of 3.

32'2*=3 3 4 *=3

Power of a power property Simplify. Because the bases are equal, the exponents are equal.

I ,c

I

x=

I

Solve forx.

01

3 O I

REFLECT\e 32 as a power of 2. 1a. Show how you can check that the solutions of the equations are correct.

1 b. In the property of equality for exponential equations, explain why b cannot be equal to 1.

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Lesson 5

1 c. How would you solve the equation in part A if 32 were replaced by 0.5?

CC.9-12AREt.1t

E X A M P L E \g an Exponential Equation with a Table The equationy = 4.1(1.33)* models the population of the United States, in millions, from 1790 to 1890. In this equation, x is the number of decades since 1790, and y is the population in millions. In what year did the population reach 45 million? A

Write an equation and make a table of values to solve the equation. When the population is 45 million, y =

.

To find the year when the population reached 45 million, solve the equation Enter the expression 4.1(1.33)* for Yj in your calculator's equation editor.

.

X .1

TflBLE SETUP

^^^^ TBL5ET_

Set up a table by pressing (E B JSQ9 and entering the values shown at TABLE right. Then press BBB BSSB to view the table.

Indpnt: Depend:

flsk Rsk

!H

Vi H.i

N.21B6 4.3106 1.1662 4.5554 4.72B4 4.B6E1

X=9

Scroll down until the value of Yj is approximately 45. When Yj is approximately 45, x —. B

Find the year when the population reached 45 million. The population reached 45 million This is

decades after 1790.

i o

years after 1790.

So, the population reached 45 million in

.

_i>

s E

2a. The table includes the ordered pair (0.4, 4.5954). What does this ordered pair represent?

id n o -

2b. Explain why it makes sense to use an increment of 0.1 for the table and not some other increment.

2c. Explain how you can check your solution.

Chapter 4

222

Lesson 5

2d. How could you solve the equation by using your calculator to graph y = 4.1(1.33)*?

Solving an Exponential Equation by Graphing Camilla invested $300 at 4% interest compounded continuously. Diego invested $275 at 6% interest compounded continuously. When will they have the same amount in their accounts? What will the amount be when this occurs? A

Write equations to represent the amount in each account. Use the fact that when an amount P is invested in an account that earns interest at a nominal rate r compounded n times per year, the amount in the account after fyears is A(i) - Pert. Camilla: A(i) =

Substitute 300 for P and 0.04

-e

for r.

Diego: A(f) —

Substitute 275 for P and 0.06

•e

for r.

j

Graph the equations. Enter the equation for Camilla's account as Yl in your calculator's equation editor. Enter the equation for Diego's account as Y2. Graph both equations in the same viewing window. A good viewing in this situation is 0 < x < 10 with a tick mark every 1 unit and 0 < y < 500 with a tick mark every 50 units.

o u

C

Find the p oint of intersection of the graphs . _______ CALC

Press CS9 BSSil and select 5:intersect to find the point of intersection of the graphs. The point of intersection is approximately So, Camilla and Diego will have the same amount in their accounts after approximately _

years.

At this time, the amount in each account will be

__

3a. Who has more money in his or her account after 3 years? How can you tell from the graphs?

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Lesson 5

3b. Suppose Camilla and Diego leave their money in their accounts for 10 years. At that time, who will have more money in his or her account? How much more?

3c. How can you observe the difference in the accounts after 10 years from the graphs of the equations?

You know that you can sometimes solve an exponential equation by writing both sides as powers with the same base. When that method is not possible, you can take a logarithm of both sides of the equation. This is justified by the following property.

Property of Equality for Logarithmic Equations For any positive numbers x, y, and b (b £ I), log^x = logfcy if and only

CC.9-12.F.BF.5(+)

EB E X A M P L E

Taking the Common Logarithm of Both Sides

Solve 2*~3 = 85. Give the exact solution and an approximate solution to three decimal places. Original equation Take the common logarithm of both sides.

log 2 = log 85

log 2

log 85

_ log 85 ~~ log 2

log 2+3 x= l^

Power Property of Logarithms

Divide both sides by log 2.

o c. (a

Simplify. Solve for x to find the exact solution.

E

Evaluate. Round to three decimal places. ID

n o

REFLECT

•

4a. Why do you use the Power Property of Logarithms?

4b. How can you use estimation to check if your answer is reasonable?

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224

Lesson 5

You can also take the natural logarithm of both sides of an equation. It makes sense to take the natural logarithm, rather than the common logarithm, when the base is e. CC.9-12.FLE.4

E X A M P L E \g the Natural Logarithm of Both Sides Adam has $500 to invest for 4 years. He wants to double his money during this time. What interest rate does Adam need for this investment, assuming the interest is compounded continuously? A

Write an equation. The formula for interest compounded continuously is A = Pert where A is the amount in the account, P is the principal, r is the annual rate of interest, and t is the time in years. P=

and A is the final amount after t = 4 years, so A =

The equation is B

Solve the equation for r. Write the equation. Divide both sides by 500.

In 2 = In e4r

Take the natural logarithm of both sides.

In 2 =

Power Property of Logarithms

In 2 =

Use the fact that In e= 1.

In2 = _

CL

4

4

Divide both sides by 4.

8

.= r

t

*•*-' /

Solve for rto find the exact answer. Evaluate. Round to three decimal places.

o vj nj I

c:

So, Adam needs an interest rate of approximately. REFLECT

o I

5a. What is the benefit of taking the natural logarithm of both sides of the equation, rather than the common logarithm?

5b. Describe two different ways to use your calculator to check your answer.

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225

Lesson 5

To solve a logarithmic equation in the form loghx = a, first rewrite the equation in exponential form (ba = x] by using the definition of a logarithm. As you will see in the second part of the following example, you may first need to isolate the logarithmic expression on one side of the equation. CC.9-12.F.BF.5(+)

EXAMPLE

Solving a Logarithmic Equation Algebraically

Solve each logarithmic equation. A

Iog3(x + 1) = 2 2

B

= x+l

Definition of logarithm

= jc+l

Simplify.

=x

Solve for x.

7 + Iog3(5:c - 4) = 10 Iog3(5x — 4) =

Subtract 7 from both sides.

3

f

= 5x — 4

Definition of logarithm

= 5x — 4

Simplify.

= 5x

Add 4 to both sides.

= x

Solve for x.

6a. How can you check your solution to part A by substitution?

6b. Your calculator has keys for evaluating only logarithms with a base of 10 or e. Use the Change of Base Property to rewrite the equation from part A so that the base of the logarithm is 10 or e. Then explain how to use graphing to check your solution.

6c. Explain how you could use graphing to check your solution to part B.

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Lesson 5

o

CC.9-12:A.REI.t1

E X A M P L E \g a Logarithmic Equation by Graphing

A telescope's limiting magnitude m is the brightness of the faintest star that can be seen using the telescope. The limiting magnitude depends on the diameter d (in millimeters) of the telescope's objective lens. The table gives two formulas relating m to d. One is a standard formula used in astronomy. The other is a proposed new formula based on data gathered from users of telescopes of various lens diameters.

Formulas for determining limiting magnitude from lens diameter Standard formula

m = 2.7 + 5 log d

Proposed formula

m = 4.5 + 4.4 log d

For what lens diameter do the two formulas give the same limiting magnitude? Use a graphing calculator. Enter 2.7 + 5 log x as Yx and enter 4.5 + 4.4 log x as Y2.

Noll Ploli

WiB2.7+51o9(X) WjB4.5+4.41o9CX

Graph the two functions in the same viewing window. Use a window where 0 < x < 2000 with a tick mark every 100 units and 0 < y < 20 with a tick mark every 5 units. CALC_

Press 69 CH and choose 5:intersect tofindthe point of intersection of the graphs. The coordinates of the point of intersection are So, the two formulas give the same limiting magnitude for a lens diameter of _

7a. What is the limiting magnitude that corresponds to this lens diameter? How do you know? a o u 01

7b. What equation can you write in order to solve the problem algebraically?

7c. Show how to solve the equation algebraically. Justify each step. (Hint: First get all logarithmic expressions on one side of the equation and all non-logarithmic expressions on the other side of the equation.) o i

Chapter 4

227

Lesson 5

PRACTICE Solve each exponential equation algebraically. 1. 16

X+2

=64

, f2f + 7 _ f 4 1 l- IsJ ~ 125;

O

4.

5. 0.01*+1 = 1000* ~ 9

7. 6* = 368

8. 0.755

3. 27X =

6 - 625 = (iF+3

10. Showthatyou can solve ^ = 16*+5 by writing both sides of the equation with a base of 2 or with a base of 4.

11. The equation}/ — 87.3(1. 07)* models the population of a city, in thousands, from 1980 to 2010. In this equation, x is the number of years since 1980, and y is the population in thousands. a. In what year did the population reach 150,000? b. In what year did the population reach 250,000? I o c

12. In the lower stratosphere (between 36,152 feet and 82,345 feet), the equation p = 473.le1'73 ~~ °-000048'! represents the atmospheric pressure p in pounds per square foot at altitude h feet.

ID 3"

a. At what altitude does the pressure equal 150 lb/ft2? b. At what altitude does the pressure equal 300 lb/ft2? 13. Rima and Trevor both bought a car in 2010. Rima's car cost $17,832 and Trevor's car cost $22,575. Rima's car is depreciating at a rate of 11% per year and Trevor's car is depreciating at a rate of 13.5% per year.

ID

8 3

a. Write each car's value as a function of time t (in years since 2010).

b. During what year will the cars have an equal value? At that time, what will the value of the cars be?

Chapter 4

228

Lesson 5

Solve. Give the exact solution and an approximate solution to three decimal places. 14. 6* =15

15. 4 2a '=200

16. 10* =35

17. 10 + e3 = 4270

18. 2 9 ~ * + 3 = 62

19.

20. 32x~1-l4

21. 210 + 4* = 3 • 4*

22. 111-X = ,

23. What happens if you take the common logarithm of both sides of 5X — —6 in order to solve the equation? Why does this happen?

24. Kendra wants to double her investment of $4000. How long will this take if the annual interest rate is 4% compounded continuously? How long will this take if the annual interest rate is 8% compounded continuously? What effect does doubling the interest rate have on the time it takes the investment to double?

e 01 c | "§ | | "Jj o

i

= J

25. An account that earns interest at an annual rate of r earns more interest each year if the account is compounded, say ntimesper year (at a rate of r/n), than if it is compounded annually. The actual interest rate .R earned is called the effective rate and r is called the nominal rate. For interest that is compounded continuously, R is given by R = er — 1. What is the nominal interest rate if R is 5.625%? Round to the nearest hundredth of a percent.

26. The equation y — 4.1(1.33)*models the population of the U.S., in millions, from 1790 to 1890. In this equation, xis the number of decades since 1790, and y is the population in millions. How many decades after 1790 did the population reach 28 million? Write an expression for the exact answer and give an approximate answer to the nearest tenth.

Chapter 4

229

Lesson 5

27. Error Analysis Identify and correct the error in the student work shown at right.

10* = 20 In 10*:= In 20 xln 10 = In 20

Solve each logarithmic equation. Round to three decimal places if necessary. 28. log? (jc - 5) = 2

29. Iog4(8x) = 3

30. log (7x - 1) = -1

31. ln(4x- 1) = 9

32.

33. 3 = In (3x + 3)

Solve by using the Product or Quotient Property of Logarithms so that one side is a single logarithm. Round to three decimal places if necessary.

34. log 20 + log I0x = 5

35. I n x - l n 6 =

36. 2.4 = log7 + log3x

For Exercises 37 and 38, use graphing to solve. 37. Charles collected data on the atmospheric pressure (ranging from 4 to 15 pounds per square inch) and the corresponding altitude above the surface of Earth (ranging from 1 to 30,000 feet). He used regression to write two functions that give the altitude above the surface of Earth given the atmospheric pressure. f(x) = 66,990 - 24,747rnx g(x) = -2870x + 40,393

IEo

a. At what atmospheric pressure(s) do the equations give the same altitude?

b. At what altitude(s) above Earth do these atmospheric pressures occur?

38. Elena and Paul determined slightly different equations to model the recommended height, in inches, of a tabletop for children x years old. I '

8

Elena: y = 12.2 + 5.45 Inx

-c

Paul:j/= 12.5 + 5.21nx For what age do the models give the same tabletop height? What is that height?

Chapter 4

230

Lesson 5

.Class.

Name.

Date-

Additional Practice Solve and check. 1. 5 2x =20

3.

KAx+7 4. 165* s* = 64

6. 25* =125x - 2

02x

2x

V

x-6

8.I-I =64

7. |

9.I-I =27

^-

Solve. 10. Iog4x5 =

11. Iog3x6 =

12. log 4 (x-6) 3 =

13. log x-log 10= 14

14. log x+ log 5 = 2

15. log(x + 9) =log(2x-7)

16. log (x + 4)- log 6 = 1

17. log x 2 ^ log 25 = 2

18. log (x- 1)2 =log (-5x- 1)

Use a table and graph to solve. 19. 2 * - 5 < 6 4

20. log x3 = 12

21. 2 X 3 X =1296

I 3 Solve. 22. The population of a small farming community is declining at a rate of 7% per year. The decline can be expressed by the exponential equation P = C (1 - 0.07)', where P is the population after t years and C is the current population. If the population was 8,500 in 2004, when will the population be less than 6,000? o

X

Chapter 4

231

Lesson 5

Problem Solving While John and Cody play their favorite video game, John drinks 4 cups of coffee and a cola, and Cody drinks 2 cups of brewed tea and a cup of iced tea. John recalls reading that up to 300 mg of caffeine is considered a moderate level of consumption per day. The rate at which caffeine is eliminated from the bloodstream is about 15% per hour. 1. John wants to know how long it will take for the caffeine in his bloodstream to drop to a moderate level. a. How much caffeine did John consume?

Caffeine Content of Some Beverages Beverage

Caffeine (mg per serving)

Brewed coffee

103

Brewed tea

36

Iced tea

30

Cola

25

b. Write an equation showing the amount of caffeine in the bloodstream as a function of time. c. How long, to the nearest tenth of an hour, will it take for the caffeine in John's system to reach a moderate level? 2. a. Cody thinks that it will take at least 8 hours for the level of caffeine in John's system to drop to the same level of caffeine that Cody consumed. Explain how he can use his graphing calculator to prove that.

b. What equations did Cody enter into his calculator?

500

y

400

c. Sketch the resulting graph.

300

200

100

Choose the letter for the best answer.

u

3. About how long would it take for the level of caffeine in Cody's system to drop by a factor of 2?

1

3

3

4

5

6

7

8

91 0

4. If John drank 6 cups of coffee and a cola, about how long would it take for the level of caffeine in his system to drop to a moderate level?

A 0.2 hour B 1.6 hours

F 0.5 hour

C 2.7 hours

G 1.6 hours

D 4.3 hours

H 4.7 hours J 5.3 hours

Chapter 4

232

Lesson 5