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CHAPTER 4
Exponential and Logarithmic Functions
We may write the Change of Base
Formula as
logb x a
Change of Base Formula
1
b loga x
loga b
logb x So, logb x is just a constant multiple
1
of loga x; the constant is
.
loga b
loga x
loga b
In particular, if we put x a, then loga a 1 and this formula becomes
logb a 1
loga b
We can now evaluate a logarithm to any base by using the Change of Base Formula to express the logarithm in terms of common logarithms or natural logarithms
and then using a calculator.
Example 6
Evaluating Logarithms with the
Change of Base Formula
Use the Change of Base Formula and common or natural logarithms to evaluate
each logarithm, correct to five decimal places.
(a) log 8 5
(b) log 9 20
We get the same answer whether we
use log10 or ln:
ln 5
⬇ 0.77398
log 8 5 ln 8
Solution
(a) We use the Change of Base Formula with b 8 and a 10:
log 8 5 log10 5
⬇ 0.77398
log10 8
(b) We use the Change of Base Formula with b 9 and a e:
log 9 20 2
Example 7
ln 20
⬇ 1.36342
ln 9
■
Using the Change of Base Formula
to Graph a Logarithmic Function
Use a graphing calculator to graph f1x2 log 6 x.
0
36
Solution Calculators don’t have a key for log6, so we use the Change of Base
Formula to write
_1
f1x2 log 6 x Figure 1
f 1x2 log 6 x 4.3
1–12
■
ln x
ln 6
Since calculators do have an LN key, we can enter this new form of the function
and graph it. The graph is shown in Figure 1.
ln x
ln 6
Exercises
Evaluate the expression.
1. log 3 127
3. log 4 log 25
5. log 4 192 log 4 3
2. log 2 160 log 2 5
1
4. log
11000
7. log 2 6 log 2 15 log 2 20
8. log 3 100 log 3 18 log 3 50
6. log 12 9 log 12 16
■
SECTION 4.3
9. log 4 16100
10,000
11. log1log 10
13–38
■
10. log 2 8 33
2
e200
12. ln1ln e
Use the Laws of Logarithms to expand the expression.
13. log 2 12x 2
14. log 3 15y 2
15. log 2 1x1x 1 22
16. log 5
17. log 610
18. ln 1 z
19. log 2 1AB 2
21. log 3 1x 1y2
24. loga a
z6
b
x1x 2 1 2
2x 2 1
y
b
31. ln a x
Bz
4
33. log 2x y
2
b
log e 2
61. Show that ln1x 2x 2 12 ln1x 2x 2 12 .
3x 2
1x 12 10
34. log a
x
3
1
1x
b
36. log 3x2y 1z
x 3 1x 1
b
3x 4
38. log a
2
10x
b
x1x 1 2 1x 4 2 2
2
40. log 12 12 log 7 log 2
41. log 2 A log 2 B 2 log 2 C
42. log 5 1x 2 1 2 log 5 1x 1 2
43. 4 log x 13 log1x 2 1 2 2 log1x 1 2
44. ln1a b2 ln1a b2 2 ln c
45. ln 5 2 ln x 3 ln1x 2 5 2
46. 21log5 x 2 log5 y 3 log5 z 2
log12x 1 2 12 3log1x 4 2 log1x 4 x 2 1 2 4
48. loga b c loga d r loga s
49–56 ■ Use the Change of Base Formula and a calculator to
evaluate the logarithm, correct to six decimal places. Use either
natural or common logarithms.
49. log 2 5
50. log 5 2
51. log 3 16
52. log 6 92
1
ln 10
60. Simplify: 1log 2 52 1log 5 72
x1
30. log 5
Bx 1
32. ln
ln x
ln 3
59. Use the Change of Base Formula to show that
a2
b
28. log a 4
b 1c
39. log 3 5 5 log 3 2
1
3
57. Use the Change of Base Formula to show that
2
39–48 ■ Use the Laws of Logarithms to combine the
expression.
47.
56. log 12 2.5
58. Draw graphs of the family of functions y loga x for
a 2, e, 5, and 10 on the same screen, using the
viewing rectangle 30, 54 by 33, 34. How are these graphs
related?
x2 4
B 1x 1 2 1x 3 7 2 2
37. ln a
55. log 4 125
26. ln 23r s
3 4
x y
29. log 2 a
54. log 6 532
357
Then use this fact to draw the graph of the function
f 1x2 log3 x.
x2
b
yz 3
3
25. ln 1ab
27. log a
x
2
22. log 2 1xy 2 10
3
23. log 5 2x 2 1
53. log 7 2.61
log 3 x 4
20. log 6 117
2
35. log
2
Laws of Logarithms
Applications
62. Forgetting Use the Ebbinghaus Forgetting Law
(Example 5) to estimate a student’s score on a biology
test two years after he got a score of 80 on a test covering
the same material. Assume c 0.3 and t is measured in
months.
63. Wealth Distribution Vilfredo Pareto (1848–1923)
observed that most of the wealth of a country is owned
by a few members of the population. Pareto’s Principle is
log P log c k log W
where W is the wealth level (how much money a person has)
and P is the number of people in the population having that
much money.
(a) Solve the equation for P.
(b) Assume k 2.1, c 8000, and W is measured in
millions of dollars. Use part (a) to find the number of
people who have $2 million or more. How many people
have $10 million or more?
64. Biodiversity Some biologists model the number of
species S in a fixed area A (such as an island) by the
Species-Area relationship
log S log c k log A
where c and k are positive constants that depend on the type
of species and habitat.
(a) Solve the equation for S.