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252
chapter 3 Exponents and Logarithms
In the era before calculators and computers existed, books of common
logarithm tables were frequently used to compute powers of numbers. As an
example of how this worked, consider how these books of logarithms would
have been used to evaluate 1.73.7 . The key to performing this calculation is
the formula
log 1.73.7 = 3.7 log 1.7.
With the advent of
calculators and computers, books of logarithms have essentially disappeared.
However, your calculator is using logarithms
and the formula
logb y t = t logb y
when you ask it to
evaluate an expression such as 1.73.7 .
Let’s assume that we have a book that gives the logarithms of the numbers
from 1 to 10 in increments of 0.001, meaning that the book gives the logarithms of 1.001, 1.002, 1.003, and so on.
The idea is first to compute the right side of the equation above. To do
that, we would look in the book of logarithms, getting log 1.7 ≈ 0.230449.
Multiplying the last number by 3.7, we would conclude that the right side
of the equation above is approximately 0.852661. Thus, according to the
equation above, we have
log 1.73.7 ≈ 0.852661.
Hence we can evaluate 1.73.7 by finding a number whose logarithm equals
0.852661. To do this, we would look through our book of logarithms and
find that the closest match is provided by the entry showing that log 7.123 ≈
0.852663. Thus 1.73.7 ≈ 7.123.
Although nowadays logarithms rarely are used directly by humans for
computations such as evaluating 1.73.7 , logarithms are used by your calculator for such computations. Logarithms also have important uses in calculus
and several other branches of mathematics. Furthermore, logarithms have
several practical uses—we will see some examples later in this chapter.
exercises
1.
For x = 7 and y = 13, evaluate each of the
following:
(a) log(x + y)
(b) log x + log y
[This exercise and the next one emphasize that
log(x + y) does not equal log x + log y.]
2.
5.
(a) log(xy)
(b) (log x)(log y)
[This exercise and the next one emphasize that
log(xy) does not equal (log x)(log y).]
(b) (log x)(log y)
For x = 12 and y = 2, evaluate each of the
following:
x
log x
(a) log y
(b)
log y
[This exercise and the next one emphasize that
log x
x
log y does not equal log y .]
(b) log x + log y
For x = 3 and y = 8, evaluate each of the
following:
For x = 1.1 and y = 5, evaluate each of the
following:
(a) log(xy)
For x = 0.4 and y = 3.5, evaluate each of the
following:
(a) log(x + y)
3.
4.
6.
For x = 18 and y = 0.3, evaluate each of the
following:
x
log x
(a) log y
(b)
log y
section 3.3 Algebraic Properties of Logarithms 253
7.
For x = 5 and y = 2, evaluate each of the
following:
(a) log x y
(b) (log x)y
[This exercise and the next one emphasize that
log x y does not equal (log x)y .]
8.
For x = 2 and y = 3, evaluate each of the
following:
For Exercises 33–40, find all numbers x that
satisfy the given equation.
33. log7 (x + 5) − log7 (x − 1) = 2
34. log4 (x + 4) − log4 (x − 2) = 3
35. log3 (x + 5) + log3 (x − 1) = 2
10. Suppose k is a positive integer such that
log k ≈ 83.2. How many digits does k have?
36. log5 (x + 4) + log5 (x + 2) = 2
log6 (15x)
=2
37.
log6 (5x)
log9 (13x)
=2
38.
log9 (4x)
39.
log(3x) log x = 4
40.
log(6x) log x = 5
11. Suppose m and n are positive integers such
that log m ≈ 32.1 and log n ≈ 7.3. How many
digits does mn have?
For Exercises 41–44, find the number of digits in
the given number.
(a) log x y
(b) (log x)y
9. Suppose N is a positive integer such that
log N ≈ 35.4. How many digits does N have?
12. Suppose m and n are positive integers such
that log m ≈ 41.3 and log n ≈ 12.8. How many
digits does mn have?
13. Suppose m is a positive integer such that
log m ≈ 13.2. How many digits does m3 have?
14. Suppose M is a positive integer such that
log M ≈ 50.3. How many digits does M 4 have?
15. Suppose log a = 118.7 and log b = 119.7. Evalub
ate a .
16. Suppose log a = 203.4 and log b = 205.4. Evalub
ate a .
17. Suppose y is such that log2 y = 17.67. Evaluate
log2 y 100 .
18. Suppose x is such that log6 x = 23.41. Evaluate
log6 x 10 .
For Exercises 19–32, evaluate the given quantities
assuming that
log3 x = 5.3 and
log3 y = 2.1,
log4 u = 3.2 and
log4 ν = 1.3.
19. log3 (9xy)
20. log4 (2uν)
x
3y
u
log4 8ν
21. log3
22.
23. log3
24. log4
√
√
x
u
1
25. log3 √
y
1
26. log4 √
ν
27. log3 (x 2 y 3 )
3
41.
74000
43.
6700 · 231000
42.
84444
44.
5999 · 172222
45.
Find an integer k such that 18k has 357
digits.
46.
Find an integer n such that 22n has 222
digits.
47.
Find an integer m such that m1234 has 1991
digits.
48.
Find an integer N such that N 4321 has 6041
digits.
49.
Find the smallest integer n such that
7n > 10100 .
50.
Find the smallest integer k such that
9k > 101000 .
51.
Find the smallest integer M such that
51/M < 1.01.
52.
Find the smallest integer m such that
81/m < 1.001.
53.
Suppose log8 (log7 m) = 5. How many digits
does m have?
54.
Suppose log5 (log9 m) = 6. How many digits
does m have?
55.
At the end of 2004, the largest known prime
number was 224036583 −1. How many digits does
this prime number have?
[A prime number is an integer greater than 1
that has no divisors other than itself and 1.]
56.
At the end of 2005, the largest known prime
number was 230402457 −1. How many digits does
this prime number have?
4
28. log4 (u ν )
29. log3
x3
y2
u2
ν3
31. log9 x 10
30. log4
32. log2 u100
254
chapter 3 Exponents and Logarithms
problems
[Sometimes seeing an alternative derivation can
help increase your understanding.]
57. Explain why
1 + log x = log(10x)
for every positive number x.
58. Explain why
2 − log x = log
100
x
for every positive number x.
59. Explain why
(1 + log x)2 = log(10x 2 ) + (log x)2
65. Derive the formula logb y1 = − logb y directly
from the formula 1/bt = b−t .
66. Without doing any calculations, explain why
the solutions to the equations in Exercises 37
and 38 are unchanged if we change the base
for all the logarithms in those exercises to any
positive number b = 1.
67.
for every positive number x.
60. Explain why
√
1 + log x
= log 10x
2
for every positive number x.
61. Pretend that you are living in the time before
calculators and computers existed, and that
you have a book showing the logarithms of
1.001, 1.002, 1.003, and so on, up to the logarithm of 9.999. Explain how you would find the
logarithm of 457.2, which is beyond the range
of your book.
62. Explain why books of logarithm tables, which
were frequently used before the era of calculators and computers, gave logarithms only for
numbers between 1 and 10.
63.
Explain why there does not exist an integer
m such that 67m has 9236 digits.
64. Derive the formula for the logarithm of a quotient by applying the formula for the logarithm
x
of a product to logb (y · y ).
Do a web search to find the largest currently
known prime number. Then calculate the number of digits in this number.
[The discovery of a new largest known prime
number usually gets some newspaper coverage, including a statement of the number of digits. Thus you can probably find on the web the
number of digits in the largest currently known
prime number; you are asked here to do the calculation to verify that the reported number of
digits is correct.]
68. Explain why expressing a large positive integer
in binary notation (base 2) should take approximately 3.3 times as many digits as expressing
the same positive integer in standard decimal
notation (base 10).
[For example, this problem predicts that
5 × 1012 , which requires 13 digits to express in
decimal notation, should require approximately
13 × 3.3 digits (which equals 42.9 digits) to express in binary notation. Expressing 5 × 1012 in
binary notation actually requires 43 digits.]
worked-out solutions to Odd-numbered Exercises
1.
For x = 7 and y = 13, evaluate each of the
following:
(a) log(x + y)
(b)
log 7 + log 13 ≈ 0.845098 + 1.113943
= 1.959041
(b) log x + log y
solution
(a) log(7 + 13) = log 20 ≈ 1.30103
3.
For x = 3 and y = 8, evaluate each of the
following:
(a) log(xy)
(b) (log x)(log y)