Chapter 4.5

Exponential and Logarithm Functions

Exponential Equations

We solved exponential equations in earlier sections. General methods for solving these equations depend on the property below, which follows from the fact that lorarithmic functions are one-to-one.

Solve 7x = 12. Give the solution to four decimal places.

127 x

12ln 7ln x

)2770.112ln

7ln x

12ln 7ln x

CautionBe careful when evaluating a quotient like

12ln

7ln

12

7ln

12ln

7ln

12ln - 7ln12ln

7ln

Solve 32x-1 = .4x+2 Give the solution to four decimal places.

212 4.3 xx

212 4.ln 3ln xx

4.ln )2(31)ln -(2x x

3ln .4ln 24.ln 3ln 2x x

Solve 32x-1 = .4x+2 Give the solution to four decimal places.

3ln .4ln 24.ln 3ln 2x x

3ln .4ln 24.ln 3ln 2 x

4.ln 3ln 2

3ln .4ln 2

x4.ln 3ln

3ln .4ln 2

2

4.ln 9ln

3ln .16ln

.49ln

3 .16ln

22.5ln

48.ln 3018.2

Solve the equationGive the solution to four decimal places.

200e2x

200 lneln 2x

200 lneln x2

eln

200 lnx2 200ln

Solve the equationGive the solution to four decimal places.

200ln 2 x

200ln x

3018.2x

Solve the equationGive the solution to four decimal places.

3eee 4x12x

3ee 4x-12x

3ee 1-2x

3eee 1-2x

e

3e

e

ee 1-2x

Solve the equationGive the solution to four decimal places.

3e 2x

3 ln)ln(e 2x

3ln eln 2x -

eln

3ln 2x - 3ln

ln32

1x 5493.

Logarithmic Equations

The next examples show some ways to solve logarithmic equations.

Solve xaaa log2)(xlog - 6)(xlog

xaa log 2)(x

6)(xlog

x

2)(x

6)(x

)2( 6 xxx

)2( 6 xxx

2xx 6x 2

62x0 2 xx

06x2 x

023x x

3x 2x

Logarithmic Equations

The negative solution x = -3 is not in the domain of logax in the original equation, so the only valid solution is the positive number 2, giving the solution set {2}.

1 1)(x log2)(3x log Solve

10 log 1)(x log2)(3x log

10 log 1)2)(x(3x log

101x 23x

1023 2 xx

Solve1023 2 xx

32

12341)1( 2 x

6

14411

6

1451

Logarithmic Equations

The number is negative,

so x-1 is negative.

So log (x-1) is not defined and this solution is not in the domain.

The solution set is

6

1451

6

1451

2ln 3xln eln lnx Solve

2ln 3xln ln x

2ln 3x

xln

2 3x

x

3)-2(x x

6-2x x

6-x-2x 0

6x

The strength of a habit is a function of the number of times the habit is repeated.If N is the number of repetitions and H is the strength of the habit, then, according to psychologist C. L. Hull

where k Is a constant .

)e(1 1000H kN

)e(1 1000H kN

Solve this equation for k.

1000

He1 kN

11000

He kN

Solve this equation for k.

11000

He kN

1000

H1e kN

)1000

H1ln()ln(e kN

Solve this equation for k.

)1000

H1ln()ln(e kN

)1000

H1ln( kN

)1000

H1ln(

1

Nk

The table gives U. S. coal consumption (in quadrillions of British thermal units, or quads) for several years. The data can be modeled with the functions defined by

where t is the number of years after 1900, and f(t) in quads.

80, t114.36, ln t 29.64 f(t)

Approximately what amount of coal was consumed in the United States in 1993?

80, t114.36, ln t 29.64 f(t)

114.36 93ln 29.64 f(93)

99.19

If this trend continues, approximately when will annual consumption reach 25 quads?

80, t114.36, ln t 29.64 f(t)

114.36 ln t 29.64 25

ln t 29.64 139.36

29.64

139.36ln t 67017543859.4

5964.70175438 ln t 5964.70175438e t

110 t

If this trend continues, approximately when will annual consumption reach 25 quads?

Annual consumption will reach 25 quads in the year 2010.

20101101900