Exponential and Logarithmic Equations and...

Name. Class- Date

Exponential and Logarithmic Equationsand Inequalities

Essential question: What is the general process for solving exponential andlogarithmic equations?

An exponential equation is an equation in which the variable appears only as anexponent. The following property is useful for solving some types of exponentialequations.

Property of Equality for Exponential Equations

For any positive number b other than l,ifbx=by, then x = y.

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

Solve each exponential equation.

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B _ 27*+!

x+ 1

32 '2*=3

34*=3

x =

REFLECT\e 32 as a power of 2.

Because the bases are equal, the exponents are equal.

Solve for x.

Write both bases as powers with a base of 3.

Power of a power property

Simplify.

Because the bases are equal, the exponents are equal.

Solve forx.

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 beequal to 1.

Chapter 4 221 Lesson 5

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

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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 isthe 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 populationreached 45 million, solve the equation .

Enter the expression 4.1(1.33)* forYj in your calculator's equation editor.

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Set up a table by pressing (E B JSQ9and entering the values shown at

TABLEright. Then press BBB BSSB to viewthe table.

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X=9

Scroll down until the valueof 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 decades after 1790.

This is years after 1790.

So, the population reached 45 million in .

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

pair represent?

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

2c. Explain how you can check your solution.

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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 $275at 6% interest compounded continuously. When will they have the same amount intheir accounts? What will the amount be when this occurs?

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

Camilla: A(i) = -e

Diego: A(f) — • e

Substitute 300 for P and 0.04for r.

Substitute 275 for P and 0.06for 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 thissituation is 0 < x < 10 with a tick mark every 1 unit and 0 < y < 500 with a tickmark every 50 units.

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 fromthe graphs?


3b. Suppose Camilla and Diego leave their money in their accounts for 10 years. At thattime, 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 ofthe equations?

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

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

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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 threedecimal places.

log 2 = log 85

Original equation

Take the common logarithm of both sides.

Power Property of Logarithms

log 2 log 85

_ log 85~~ log 2

x=l^ + 3log 2

Divide both sides by log 2.

Simplify.

Solve for x to find the exact solution.

Evaluate. Round to three decimal places.

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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You can also take the natural logarithm of both sides of an equation. It makes sense to takethe natural logarithm, rather than the common logarithm, when the base is e.

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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 thistime. What interest rate does Adam need for this investment, assuming theinterest is compounded continuously?

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A Write an equation.

The formula for interest compounded continuously is A = Pert where A is the amount inthe 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.

In 2 = In e4r

In 2 =

In 2 =

In2 = _4 4

.= r

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Write the equation.

Divide both sides by 500.

Take the natural logarithm of both sides.

Power Property of Logarithms

Use the fact that In e= 1.

Divide both sides by 4.

Solve for rto find the exact answer.

Evaluate. Round to three decimal places.

So, Adam needs an interest rate of approximately.

REFLECT

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.


To solve a logarithmic equation in the form loghx = a, first rewrite the equation inexponential form (ba = x] by using the definition of a logarithm. As you will see in thesecond part of the following example, you may first need to isolate the logarithmicexpression on one side of the equation.

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E X A M P L E Solving a Logarithmic Equation Algebraically

Solve each logarithmic equation.

A Iog3(x + 1) = 22= x+l Definition of logarithm

= jc+l Simplify.

= x Solve for x.

B 7 + Iog3(5:c - 4) = 10

Iog3(5x — 4) = Subtract 7 from both sides.

3

= 5x — 4 Definition of logarithm

= 5x — 4 Simplify.

= 5x Add 4 to both sides.

= x Solve for x.

f 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. Usethe Change of Base Property to rewrite the equation from part A so that the base ofthe 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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E X A M P L E \g a Logarithmic Equation by Graphing

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A telescope's limiting magnitude m is the brightness ofthe faintest star that can be seen using the telescope.The limiting magnitude depends on the diameter d (inmillimeters) of the telescope's objective lens. The tablegives two formulas relating m to d. One is a standardformula used in astronomy. The other is a proposed newformula based on data gathered from users of telescopesof various lens diameters.

For what lens diameter do the two formulas give the same limiting magnitude?

Formulas for determining limitingmagnitude from lens diameter

Standardformula

Proposedformula

m = 2.7 + 5 log d

m = 4.5 + 4.4 log d

Use a graphing calculator. Enter 2.7 + 5 log x as Yx and enter4.5 + 4.4 log x as Y2.

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

CALC_

Press 69 CH and choose 5:intersect to find the point ofintersection of the graphs.

The coordinates of the point of intersection are

Noll PloliWiB2.7+51o9(X)WjB4.5+4.41o9CX

So, the two formulas give the samelimiting magnitude for a lens diameter of _

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

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 getall logarithmic expressions on one side of the equation and all non-logarithmicexpressions on the other side of the equation.)


P R A C T I C E

Solve each exponential equation algebraically.

1. 16 =64X+2 , f2f + 7 _ f 4 1

l- IsJ ~ 125;O

3. 27X =

4. 5. 0.01*+1 = 1000* ~ 9 6-625 = (iF+3

7. 6* = 368 8. 0.755

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?

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

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 andTrevor's car cost $22,575. Rima's car is depreciating at a rate of 11% per yearand Trevor's car is depreciating at a rate of 13.5% per year.

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

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


Solve. Give the exact solution and an approximate solution to three decimal places.

14. 6* =15 15. 42a'=200 16. 10* =35

17. 10 + e3 = 4270 18. 29~*+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 orderto solve the equation? Why does this happen?

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

eo01 25. An account that earns interest at an annual rate of r earns more interest each

i c| year if the account is compounded, say n times per 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

"Jj 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, fromJ 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 reach28 million? Write an expression for the exact answer and give an approximateanswer to the nearest tenth.


27. Error Analysis Identify and correct the error in thestudent 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 oneside is a single logarithm. Round to three decimal places if necessary.

34. log 20 + log I0x = 5 35. Inx-ln6 = 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 poundsper 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 givethe altitude above the surface of Earth given the atmospheric pressure.

f(x) = 66,990 - 24,747rnx

g(x) = -2870x + 40,393

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

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

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

Elena: y = 12.2 + 5.45 Inx

Paul:j/= 12.5 + 5.21nx

For what age do the models give the same tabletop height? What is that height?

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Name. .Class.

Additional Practice

Date-

Solve and check.

1. 52x=20 3.

5* K A x + 74. 16s* = 64 02x 6. 25* =125x-2

7. |V ^-

2x

8.I-I =64x-6

9.I-I =27

Solve.

10. Iog4x5 = 11. Iog3x6 = 12. log4(x-6)3 =

13. log x-log 10= 14 14. log x+ log 5 = 2 15. log(x + 9) =log(2x-7)

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16. log (x + 4)- log 6 = 1 17. log x2^ log 25 = 2

Use a table and graph to solve.

19. 2 * - 5 < 6 4 20. log x3 = 12

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

21. 2X3X=1296

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 equationP = C (1 - 0.07)', where P is the population after t years and C is the currentpopulation. If the population was 8,500 in 2004, when will the population beless than 6,000?


Problem Solving

While John and Cody play their favorite video game,John drinks 4 cups of coffee and a cola, and Codydrinks 2 cups of brewed tea and a cup of iced tea.John recalls reading that up to 300 mg of caffeine isconsidered a moderate level of consumption per day.The rate at which caffeine is eliminated from thebloodstream is about 15% per hour.

1. John wants to know how long it will take for thecaffeine in his bloodstream to drop to a moderatelevel.

a. How much caffeine did John consume?

b. Write an equation showing the amount of caffeinein the bloodstream as a function of time.

c. How long, to the nearest tenth of an hour, will ittake for the caffeine in John's system to reacha moderate level?

Caffeine Content of SomeBeverages

Beverage

Brewed coffee

Brewed tea

Iced tea

Cola

Caffeine(mg perserving)

103

36

30

25

2. a. Cody thinks that it will take at least 8 hours for the level of caffeine inJohn'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 intohis calculator?

c. Sketch the resulting graph.

Choose the letter for the best answer.

3. About how long would it take for thelevel of caffeine in Cody's system todrop by a factor of 2?

A 0.2 hour

B 1.6 hours

C 2.7 hours

D 4.3 hours

500

400

300

200

100

y

u 1 3 3 4 5 6 7 8 9 1 0

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

F 0.5 hour

G 1.6 hours

H 4.7 hours

J 5.3 hours


Exponential and Logarithmic Equations and...

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