Chapter 6

Exponential and Logarithmic Functions and Applications

Section 6.4

Section 6.4 Exponential and Logarithmic Equations

• Solving Exponential Equationso Exponential Equality Propertyo Base 10 or Base eo Graphically

• Fundamental Properties of Logarithmso Solving Exponential Equations Using Properties of Logarithms

• Solving Logarithmic Equations

• Solving Literal Equations Involving Exponential and Logarithmic Equations

• Change of Base Formula

• Applications

Exponential Equality Property:

For b > 0 and b 1, if bm = bn, then m = n.

*Solving Exponential Equations with the Exponential Equality Property:

1.Express each side of the equation as a power of the same base.

2.Apply the exponential equality property to equate the exponents.

3. Solve for the variable.

4. Check your solution.

*Also referred to as solving by "equating the bases" or "relating the bases."

Use the exponential equality property to solve 292x + 4 = 162.

Step 1. Express each side of the equation as a power of the same base.

First, we divide both sides of 292x + 4 = 162 by “2.”

92x + 4 = 81

Next, we express each side of the equation as a power of “3.”

(32)2x + 4 = 34

34x + 8 = 34

Make sure to distribute correctly: 2(2x + 4) = 4x + 8

Steps 2-3. Apply the exponential equality property to equate the exponents. Solve for the variable.

4x + 8 = 4 x = –1

Step 4. Check your solution.

29[2(–1) + 4] = 162 292 = 162 2(81) = 192 True

Use the exponential equality property to solve

Check:

1x21/2 55

22x1/2 55

22x1/2

x5/4

2.23612.2361

255

2551/4

1] [(5/4)

.1x255

Solving Exponential Equations with Base 10 or Base e.

1.Isolate the power (the term containing the variable exponent) on one side of the equation.

If necessary, divide both sides of the equation by any coefficient of the power term.

2. Convert the equation to logarithmic form.

3. Solve for the variable. Use a calculator if necessary.

4. Check your solution.

Solve the exponential equation 2(–6 + 102x) = 16.26. Round your answer to 4 decimal places as needed.

Step 1. Isolate the power. First, divide both sides of the equation by “2.” Then, add “6” to both sides of the equation.

–6 + 102x = 8.13

102x = 14.13

Step 2. Convert the equation to logarithmic form. log1014.13 = 2x or log 14.13 = 2x

Step 3. Solve for the variable. x = log 14.13

2

x 0.5751

Step 4. Check your solution.

2[–6 + 10(2 0.5751)] = 16.26 2(8.1319) = 16.2638 16.26

Solve the exponential equation 25.6 = –2 + 3.1e–0.15x. Round your answer to 4 decimal places as needed.

27.6 = 3.1e–0.15x

27.6 = 3.1e–0.15x

3.1 3.1

8.9032 = e–0.15x

loge 8.9032 = –0.15x or ln 8.9032 = –0.15x

x = ln 8.9032 –0.15

x –14.5761

You can check the answer.

Recall the Basic Properties of Logarithms: For b > 0 and b 1,

logb 1 = 0, logb b = 1, logb bx = x, and

Fundamental Properties of Logarithms

For positive real numbers M, N, and b, b 1, and any real number k:

Product Property:

Quotient Property:

Power Property:

Note: These properties apply to natural logarithms as well.

x.b xlogb

NlogMlog(MN)log bbb

NlogMlogNM

log bbb

Mlog kMlog bk

b

Use the fact that log3 4 = 1.2619 and log3 7 = 1.7712 and the

properties of logarithms to estimate the value of the following expressions. Round your answers to 4 decimal places as needed.

a. log3 28

= log3 (47) = log3(4) + log3(7) = 1.2619 + 1.7712 = 3.0331

b. log3 (1.75)

= log3 (7/4) = log3(7) – log3(4) = 1.7712 – 1.2619 = 0.5093

c.

4log3

0.630919)(1/2)(1.264log 1/24log 33 21 /

Use the properties of logarithms to rewrite each expression as a single logarithm.

a. 5 log x + log 9

= log x5 + log 9 = log (9x5)

b. log (x + 3) – log (x2 – 9)

c.

or

8logy)logxlog (231

bbb

3x1

log3)3)(x(x

3x log

9x

3x log

2

8logy

xlog

31

8logy)logx(log31

b

2

bbb2

b

1/32

bb

2

b yx

8log 8logy

xlog

31 /

3

2

b yx

8log

Expand the given expression in terms of simpler logarithms. Assume that all variable expressions are positive real numbers.

yxlog

2

21 /

yx

log2

y) logx (log21 2

yx

log2

21

y) logx log (221

y log x log 21

Solving Exponential Equations Using Properties of Logarithms

1. Isolate the power on one side of the equation.

2. Take the logarithm of both sides of the equation; may take either the common logarithm (base 10) or a natural logarithm (base e).

3. Apply the power property of logarithms to simplify (that is, "bring down" the variable exponent to the front).

4. Solve for the variable.

5. Check your solution.

Solve the exponential equation –7 + 4x = 5. Round your answer to 4 decimal places.

Step 1: Isolate the power on one side of the equation. 4x = 12

Step 2: Take the logarithm of both sides of the equation. log 4x = log 12

Step 3: Apply the power property of logarithms. x log 4 = log 12

Step 4: Solve for the variable.

5. Check your solution.

79251.4 log

12 logx

–7 + 4(1.7925) 5

Solve the exponential equation 27ex – 2 = 7. Round your answer to 4 decimal places.

ex – 2 = 0.2593

ln ex – 2 = ln 0.2593

(x – 2) ln e = ln 0.2593

(x – 2) (1) = ln 0.2593

(Recall ln e is equivalent to loge e)

x = ln 0.2593 + 2

x 0.6502

Checking:

27e(0.6502 – 2) 7

Logarithmic Equality Property:

For positive real numbers m, n, and b, b 1, if logb m = logb n, then m = n.

Solving Logarithmic Equations:

1. Isolate the logarithmic expression on one side of the equation. If needed, apply the properties of logarithms to combine all logarithms as a single logarithm.

2. Convert the logarithmic equation to an exponential equation.

3. Solve for the variable.

4. Check for possible extraneous solutions.

Solve the logarithmic equation 6 log 4x = 18.

Step1. Isolate the logarithmic expression on one side of theequation. log 4x = 3

Step 2: Convert the logarithmic equation to an exponentialequation.

103 = 4x

Step 3: Solve for the variable. 4x = 1000

x = 250

Step 4: Check for possible extraneous solutions.

6 log [4(250)] = 18

18 = 18

Solve the logarithmic equation

We discard x = –10/3 (it is not in the domain). Thus, x = 10.You can verify the solution.

(3x). log2320

x log

2log(3x)320

x log

2320

x(3x) log

20x 3x 10 22

2 20x) log(3x2

010020x3x2

010)10)(x(3x

10 x or 10/3x

Solve the equation ln x + ln (x + 3) = ln (x + 15).

ln [x(x + 3)] = ln (x + 15)

ln (x2 + 3x) = ln (x + 15)

x2 + 3x = x + 15 Logarithmic equality

x2 + 3x – x – 15 = 0

x2 + 2x – 15 = 0

(x + 5)(x – 3) = 0

x = –5 or x = 3

We discard x = –5 (it is not in the domain). Thus, x = 3.You can verify the solution.

Solve the equation ln (2x – 1) = 3 and approximate your answer to 4 decimal places.

Recall ln (2x – 1) = 3 is equivalent to loge(2x – 1) = 3

e3 = 2x – 1

2x = e3 + 1

2x = 21.085537

x 10.5428

Solving Literal Equations Involving Exponential orLogarithmic Equations

Solve for y:

ylog mnxlog bb

mnylog xlog bb

mnyx

logb

yx

bmn

(y)yx

)(y)(bmn

mnb

xy

Change of Base Formula

For positive real numbers M, a, and b, a 1, b 1,

We may convert any given base into either base 10 or base e.

Example: Use the change of base formula to evaluate log2 5, and

graph y = log2 x by applying the change of base formula.

or

Graph of y = log2 x:

ba

ab log

MlogMlog

32192.2 log5 log

5 log2 32192.2 ln5 ln

5 log2

The function P(t) = 89.371(3.2)t models the number of digital 3D screens worldwide for t number of years after 2005. Using this model, estimate when the number of digital 3D screens worldwide reached approximately 9,000. Solve algebraically and round your answer to the nearest whole number. Source: mpaa.org.

The number of digital 3D screens reached approximately 9,000 in2009.

9000)89.371(3.2 t

100.70383.2t

100.7038 log3.2 log t

100.7038 log3.2 log t

43.2 log

100.7038 log t

The number of prohibited firearms intercepted by the Transportation Security Administration (TSA) at U.S. airport screenings from 2005 to 2009 can be modeled by the functionf(x) = 2391.893 – 931.691 ln x, where x = 1 represents 2005, x = 2 is 2006, and so on. Use this model to estimate when the TSA intercepted approximately 892 firearms. Round your answer to the nearest whole number. Sources: www.aaa.com; www.safecarguide.com

 

The TSA intercepted approximately 892 firearms in 2009.

892x ln 931.6912391.893

1499.893x ln 931.691

1.6099x ln

1.6099x e

5x

Using your textbook, practice the problems assigned by your instructor to review the concepts from Section 6.4.