chapter 4.5 exponential and logarithm functions. exponential equations we solved exponential...
TRANSCRIPT
![Page 1: Chapter 4.5 Exponential and Logarithm Functions. Exponential Equations We solved exponential equations in earlier sections. General methods for solving](https://reader035.vdocument.in/reader035/viewer/2022081512/56649f1c5503460f94c32487/html5/thumbnails/1.jpg)
Chapter 4.5
Exponential and Logarithm Functions
![Page 2: Chapter 4.5 Exponential and Logarithm Functions. Exponential Equations We solved exponential equations in earlier sections. General methods for solving](https://reader035.vdocument.in/reader035/viewer/2022081512/56649f1c5503460f94c32487/html5/thumbnails/2.jpg)
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.
![Page 3: Chapter 4.5 Exponential and Logarithm Functions. Exponential Equations We solved exponential equations in earlier sections. General methods for solving](https://reader035.vdocument.in/reader035/viewer/2022081512/56649f1c5503460f94c32487/html5/thumbnails/3.jpg)
![Page 4: Chapter 4.5 Exponential and Logarithm Functions. Exponential Equations We solved exponential equations in earlier sections. General methods for solving](https://reader035.vdocument.in/reader035/viewer/2022081512/56649f1c5503460f94c32487/html5/thumbnails/4.jpg)
Solve 7x = 12. Give the solution to four decimal places.
127 x
12ln 7ln x
)2770.112ln
7ln x
12ln 7ln x
![Page 5: Chapter 4.5 Exponential and Logarithm Functions. Exponential Equations We solved exponential equations in earlier sections. General methods for solving](https://reader035.vdocument.in/reader035/viewer/2022081512/56649f1c5503460f94c32487/html5/thumbnails/5.jpg)
CautionBe careful when evaluating a quotient like
12ln
7ln
12
7ln
12ln
7ln
12ln - 7ln12ln
7ln
![Page 6: Chapter 4.5 Exponential and Logarithm Functions. Exponential Equations We solved exponential equations in earlier sections. General methods for solving](https://reader035.vdocument.in/reader035/viewer/2022081512/56649f1c5503460f94c32487/html5/thumbnails/6.jpg)
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
![Page 7: Chapter 4.5 Exponential and Logarithm Functions. Exponential Equations We solved exponential equations in earlier sections. General methods for solving](https://reader035.vdocument.in/reader035/viewer/2022081512/56649f1c5503460f94c32487/html5/thumbnails/7.jpg)
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
![Page 8: Chapter 4.5 Exponential and Logarithm Functions. Exponential Equations We solved exponential equations in earlier sections. General methods for solving](https://reader035.vdocument.in/reader035/viewer/2022081512/56649f1c5503460f94c32487/html5/thumbnails/8.jpg)
Solve the equationGive the solution to four decimal places.
200e2x
200 lneln 2x
200 lneln x2
eln
200 lnx2 200ln
![Page 9: Chapter 4.5 Exponential and Logarithm Functions. Exponential Equations We solved exponential equations in earlier sections. General methods for solving](https://reader035.vdocument.in/reader035/viewer/2022081512/56649f1c5503460f94c32487/html5/thumbnails/9.jpg)
Solve the equationGive the solution to four decimal places.
200ln 2 x
200ln x
3018.2x
![Page 10: Chapter 4.5 Exponential and Logarithm Functions. Exponential Equations We solved exponential equations in earlier sections. General methods for solving](https://reader035.vdocument.in/reader035/viewer/2022081512/56649f1c5503460f94c32487/html5/thumbnails/10.jpg)
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
![Page 11: Chapter 4.5 Exponential and Logarithm Functions. Exponential Equations We solved exponential equations in earlier sections. General methods for solving](https://reader035.vdocument.in/reader035/viewer/2022081512/56649f1c5503460f94c32487/html5/thumbnails/11.jpg)
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.
![Page 12: Chapter 4.5 Exponential and Logarithm Functions. Exponential Equations We solved exponential equations in earlier sections. General methods for solving](https://reader035.vdocument.in/reader035/viewer/2022081512/56649f1c5503460f94c32487/html5/thumbnails/12.jpg)
Logarithmic Equations
The next examples show some ways to solve logarithmic equations.
![Page 13: Chapter 4.5 Exponential and Logarithm Functions. Exponential Equations We solved exponential equations in earlier sections. General methods for solving](https://reader035.vdocument.in/reader035/viewer/2022081512/56649f1c5503460f94c32487/html5/thumbnails/13.jpg)
Solve xaaa log2)(xlog - 6)(xlog
xaa log 2)(x
6)(xlog
x
2)(x
6)(x
)2( 6 xxx
![Page 14: Chapter 4.5 Exponential and Logarithm Functions. Exponential Equations We solved exponential equations in earlier sections. General methods for solving](https://reader035.vdocument.in/reader035/viewer/2022081512/56649f1c5503460f94c32487/html5/thumbnails/14.jpg)
)2( 6 xxx
2xx 6x 2
62x0 2 xx
06x2 x
023x x
3x 2x
![Page 15: Chapter 4.5 Exponential and Logarithm Functions. Exponential Equations We solved exponential equations in earlier sections. General methods for solving](https://reader035.vdocument.in/reader035/viewer/2022081512/56649f1c5503460f94c32487/html5/thumbnails/15.jpg)
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}.
![Page 16: Chapter 4.5 Exponential and Logarithm Functions. Exponential Equations We solved exponential equations in earlier sections. General methods for solving](https://reader035.vdocument.in/reader035/viewer/2022081512/56649f1c5503460f94c32487/html5/thumbnails/16.jpg)
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
![Page 17: Chapter 4.5 Exponential and Logarithm Functions. Exponential Equations We solved exponential equations in earlier sections. General methods for solving](https://reader035.vdocument.in/reader035/viewer/2022081512/56649f1c5503460f94c32487/html5/thumbnails/17.jpg)
Solve1023 2 xx
32
12341)1( 2 x
6
14411
6
1451
![Page 18: Chapter 4.5 Exponential and Logarithm Functions. Exponential Equations We solved exponential equations in earlier sections. General methods for solving](https://reader035.vdocument.in/reader035/viewer/2022081512/56649f1c5503460f94c32487/html5/thumbnails/18.jpg)
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
![Page 19: Chapter 4.5 Exponential and Logarithm Functions. Exponential Equations We solved exponential equations in earlier sections. General methods for solving](https://reader035.vdocument.in/reader035/viewer/2022081512/56649f1c5503460f94c32487/html5/thumbnails/19.jpg)
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
![Page 20: Chapter 4.5 Exponential and Logarithm Functions. Exponential Equations We solved exponential equations in earlier sections. General methods for solving](https://reader035.vdocument.in/reader035/viewer/2022081512/56649f1c5503460f94c32487/html5/thumbnails/20.jpg)
![Page 21: Chapter 4.5 Exponential and Logarithm Functions. Exponential Equations We solved exponential equations in earlier sections. General methods for solving](https://reader035.vdocument.in/reader035/viewer/2022081512/56649f1c5503460f94c32487/html5/thumbnails/21.jpg)
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
![Page 22: Chapter 4.5 Exponential and Logarithm Functions. Exponential Equations We solved exponential equations in earlier sections. General methods for solving](https://reader035.vdocument.in/reader035/viewer/2022081512/56649f1c5503460f94c32487/html5/thumbnails/22.jpg)
)e(1 1000H kN
Solve this equation for k.
1000
He1 kN
11000
He kN
![Page 23: Chapter 4.5 Exponential and Logarithm Functions. Exponential Equations We solved exponential equations in earlier sections. General methods for solving](https://reader035.vdocument.in/reader035/viewer/2022081512/56649f1c5503460f94c32487/html5/thumbnails/23.jpg)
Solve this equation for k.
11000
He kN
1000
H1e kN
)1000
H1ln()ln(e kN
![Page 24: Chapter 4.5 Exponential and Logarithm Functions. Exponential Equations We solved exponential equations in earlier sections. General methods for solving](https://reader035.vdocument.in/reader035/viewer/2022081512/56649f1c5503460f94c32487/html5/thumbnails/24.jpg)
Solve this equation for k.
)1000
H1ln()ln(e kN
)1000
H1ln( kN
)1000
H1ln(
1
Nk
![Page 25: Chapter 4.5 Exponential and Logarithm Functions. Exponential Equations We solved exponential equations in earlier sections. General methods for solving](https://reader035.vdocument.in/reader035/viewer/2022081512/56649f1c5503460f94c32487/html5/thumbnails/25.jpg)
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)
![Page 26: Chapter 4.5 Exponential and Logarithm Functions. Exponential Equations We solved exponential equations in earlier sections. General methods for solving](https://reader035.vdocument.in/reader035/viewer/2022081512/56649f1c5503460f94c32487/html5/thumbnails/26.jpg)
![Page 27: Chapter 4.5 Exponential and Logarithm Functions. Exponential Equations We solved exponential equations in earlier sections. General methods for solving](https://reader035.vdocument.in/reader035/viewer/2022081512/56649f1c5503460f94c32487/html5/thumbnails/27.jpg)
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
![Page 28: Chapter 4.5 Exponential and Logarithm Functions. Exponential Equations We solved exponential equations in earlier sections. General methods for solving](https://reader035.vdocument.in/reader035/viewer/2022081512/56649f1c5503460f94c32487/html5/thumbnails/28.jpg)
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
![Page 29: Chapter 4.5 Exponential and Logarithm Functions. Exponential Equations We solved exponential equations in earlier sections. General methods for solving](https://reader035.vdocument.in/reader035/viewer/2022081512/56649f1c5503460f94c32487/html5/thumbnails/29.jpg)
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