Exponential And Logarithmic Equations Worksheet - Section 4-7 Page 8
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4-7 Exponential and Logarithmic Equations
E x p l o r e / D i s c u s s
If b is any positive real number different from 1, the change-of-base
formula implies that the function y
log
x is a constant multiple of the
b
natural logarithmic function; that is, log
x
k ln x for some k.
2
b
(A) Graph the functions y
ln x, y
2 ln x, y
0.5 ln x, and
y
3 ln x.
(B) Write each function of part A in the form y
log
x by finding the
b
base b to two decimal places.
x
x
(C) Is every exponential function y
b
a constant multiple of y
e
?
Explain.
A n s w e r s t o M a t c h e d P r o b l e m s
1. x
0.2263
2. More than double in 9 years, but not quite double in 8 years
3. 9.9 miles
4. x
1.195
v/c
5. x
20
6. x
1,100
7. W
W
e
8. 4.7486
b
t
2
2
x
x
EXERCISE 4-7
25.
e
0.23
26.
e
125
Solve Problems 27–38 exactly.
A
27. log x
log 5
log 2
log (x
3)
Solve Problems 1–12 algebraically and check graphically.
28. log (6x
5)
log 3
log 2
log x
Round answers to three significant digits.
29. ln x
ln (2x
1)
ln (x
2)
x
x
3x 1
1. 10
0.0347
2. 10
14.3
3. 10
92
30. ln (x
1)
ln (3x
1)
ln x
5x 2
x
x
4. 10
348
5. e
3.65
6. e
0.0142
31. log (2x
1)
1
log (x
1)
2x 1
3x 5
x
7. e
405
8. e
23.8
9. 5
18
32. 1
log (x
2)
log (3x
1)
x
x
x
10. 3
4
11. 2
0.238
12. 3
0.074
3
4
3
4
33. (ln x)
ln x
34. (log x)
log x
Solve Problems 13–18 exactly.
35. ln (ln x)
1
36. log (log x)
1
13. log 5
log x
2
14. log x
log 8
1
log x
log x
37. x
100x
38. 3
3x
15. log x
log (x
3)
1
16. log (x
9)
log 100x
3
In Problems 39–40,
(A) Explain the difficulty in solving the equation exactly.
17. log (x
1)
log (x
1)
1
(B) Determine the number of solutions by graphing the
18. log (2x
1)
1
log (x
2)
functions on each side of the equation.
x/2
B
39.
e
5 ln x
40.
ln (ln x)
ln x
2
Solve Problems 19–26 algebraically and check graphically.
In Problems 41–42,
Round answers to three significant digits.
(A) Explain the difficulty in solving the equation exactly.
x
x
19. 2
1.05
20. 3
1.06
(B) Use a graphing utility to find all solutions to three decimal
1.4x
0.32x
21. e
13
22. e
632
places.
0.12x
0.25x
x
x
x/4
41.
3
2
7
x
e
42.
e
5 log x
4 ln x
23. 123
500e
24. 438
200e
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