§ 9.5 Exponential and Logarithmic Equations. Blitzer, Intermediate Algebra, 5e – Slide #2 Section 9.5 Exponential Equations At age 20, you inherit $30,000

§ 9.5

Exponential and Logarithmic Equations

Blitzer, Intermediate Algebra, 5e – Slide #2 Section 9.5

Exponential Equations

At age 20, you inherit $30,000. You’d like to put aside $25,000 and eventually have over half a million dollars for early retirement. Is this possible? In this section, you will see how techniques for solving equations with variable exponents provide an answer to this question.

One technique involves using logarithms to solve the exponentialequations. Remember that you are solving for x and that the x in these equations is in the exponent. So this is different than any equation we have looked at before.

And for the answer to the question on when the $25,000 would grow to $500,000 –If the money was invested at 9% compounded monthly, it would grow to the half million dollars you wanted for your retirement in 33.4 years.



Solving Exponential Equations by Expressing Each Side as a

Power of the Same Base. then ,1 and If NMbbb NM

Express each side as a power of the same base.

Set the exponents equal to each other.



EXAMPLEEXAMPLE

Solve:

SOLUTIONSOLUTION

.77b273a 6

212

xx

This is the given equation.

In each equation, express both sides as a power of the same base. Then set the exponents equal to each other.(a) Because 27 is , we express each side of the equation in terms of the base, 3.

123 x

273 12 x

Write each side as a power of the same base.

312 33 x

Equate the exponents.312 x

Subtract 1 from both sides.22 x



Divide both sides by 2.1x

CONTINUECONTINUEDD

Check 1:273 12 x

273 112

2733 2727

?

?


(b) Because can both be expressed as 7 raised to a power, we will write each side of the equation in terms of 7.

77 6

2

x

7 and 7 6

2x

The solution is 1 and the solution set is {1}.true



CONTINUECONTINUEDD Write each side as a power of

the same base.2

1

6

2

77 x

Equate the bases.2

1

6

2

x

Multiply both sides by the LCD, 6.

32 x

Add 2 to both sides.5x

Check 5:

77 6

2

x

77 6

25

?



CONTINUECONTINUEDD

77 6

3

?

77 2

1

?

77

The solution is 5 and the solution set is {5}.

true



Using Natural Logarithms to Solve Exponential Equations

1) Isolate the exponential expression.

2) Take the natural logarithm on both sides of the equation.

3) Simplify using one of the following properties:

4) Solve for the variable.

.lnor lnln xebxb xx



EXAMPLEEXAMPLE

Solve:

SOLUTIONSOLUTION

.62501250 065.0 xe


We begin by dividing both sides by 1250 to isolate the exponential expression, . Then we take the natural logarithm on both sides of the equation.

xe 065.0

62501250 065.0 xe

Isolate the exponential factor by dividing both sides by 1250.

5065.0 xe

Take the natural logarithm on both sides.

5lnln 065.0 xe



Use the inverse property5ln065.0 x

CONTINUECONTINUEDD

on the left side.xex ln

Divide both sides by 0.065.76.24065.0

5lnx

Thus, the solution of the equation is . Try checking

this approximate solution in the original equation, verifying that

is the solution set.

76.24065.0

5ln

76.24

065.0

5ln


Logarithmic Equations

EXAMPLEEXAMPLE

Solve:

SOLUTIONSOLUTION

.514log2 x

This is the equivalent equation.

We first rewrite the equation as an equivalent equation in exponential form using the fact that . means log xbcx c

b

.142 means 514log 52 xx

Now we solve the equivalent equation for x.

1425 xEvaluate the exponent.1432 x

Subtract 1 from both sides.x431

Divide both sides by 4.x4

31



This is the given logarithmic equation.

CONTINUECONTINUEDD

Check 31/4:

514log2 x

Replace x with 31/4.514

314log2

Multiply. 5131log2

Add. 532log2

55 .322 because 532log 52

?

?

?

true

This true statement indicates that the solution is 31/4 and the solution set is {31/4}.



EXAMPLEEXAMPLE

Solve:

SOLUTIONSOLUTION

.2log12log(b)31log1log(a) 22 xxxx

This is the given equation. 31log1log(a) 22 xx

Use the product rule to obtain a single logarithm.

311log2 xx

Multiply. 31log 22 x

.12 means 31log 2322 xx12 23 x

18 2 x Evaluate the exponent.

90 2 x Subtract 8 from both sides. 033 xx Factor.



03or 03 xx Set each factor equal to 0.

CONTINUECONTINUEDD

3x Solve for x.3x

Check 3: Check -3:

31log1log 22 xx 31log1log 22 xx

313log13log 22 313log13log 22

34log2log 22 32log4log 22

Negative numbers do not have logarithms.

false

342log2

38log2

823

??

?

?

?

?



CONTINUECONTINUEDD 88 true

The solutions are 3 and -3. The solution set is {3, -3}.

2log12log(b) xx This is the given equation.

212

log

x

x Use the quotient rule to obtain a single logarithm.

.12

10 means 212

log 2

x

x

x

x

x

x 12102

x

x 12100

Evaluate the exponent.

12100 xx Multiply both sides by the LCD, x.



CONTINUECONTINUEDD

198 x Subtract 2x from both sides.

98

1x Divide both sides by 98.

Check -1/98:

2log12log xx

298

1log1

98

12log

Negative numbers do not have logarithms. Therefore there is no solution and the solution set is .

false



EXAMPLEEXAMPLE

Solve:

SOLUTIONSOLUTION

.6ln37 x

This is the given equation.6ln37 x

Subtract 7 from both sides.1ln3 x

Divide both sides by 3.3

1ln x

Exponentiate both sides.3

1ln

ee x

Simplify the left side.3

1

ex

Evaluate the exponent.72.0x



Check :

CONTINUECONTINUEDD 3

1

e

This is the given equation.6ln37 x

Replace x with6ln37 3

1

e .3

1

e

Because , we conclude63

137

xex ln

.3

1ln 3

1

e

Multiply.617

Subtract.66

This true statement indicates that the solution is and the

solution set is

3

1

e

.3

1

e

?

?

?

true


Exponential Equations in Application

EXAMPLEEXAMPLE

The formula models the population of Florida, A, in millions, t years after 2000. When will the population of Florida reach 19.2 million?

SOLUTIONSOLUTION


teA 0235.09.15

We will replace A with 19.2 and then solve for t.teA 0235.09.15

Replace A with 19.2.te 0235.09.152.19

Divide both sides by 15.9.te 0235.02.1

Take the natural logarithm of both sides.

te 0235.0ln2.1ln


Exponential Equations in Application

Use the inverse property on the right side.

t0235.02.1ln

CONTINUECONTINUEDD

xex ln

Divide both sides by 0.0235.76.70235.0

2.1lnt

The state of Florida will have a population of 19.2 million people approximately 7.76 years after the year 2000.

Documents

§ 9.5 Exponential and Logarithmic Equations. Blitzer, Intermediate Algebra, 5e – Slide #2 Section 9.5 Exponential Equations At age 20, you inherit $30,000