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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
