MAC 1140
More practice with Log Properties
Section 4.3
Additional Homework
In exercises 1 – 10, suppose b is a positive constant greater than 1, and let A, B, and C be
logb 3 B
logb 2 A
logb 5 C
defined as follows:
In each case, use the properties of logarithms to evaluate the given expression in terms of
A, B, and/or C. (In exercises 5 – 10, use the change-of-base formula.)
1. (a) logb 6
(b) logb (1/ 6)
(c) logb 27
(d) logb (1/ 27)
2. (a) logb 10
(b) logb 100
(c) logb 0.01
(d) logb 0.3
3. (a) logb (5 / 3)
(b) logb 0.6
(c) logb (5 / 9)
(d) logb (5 /16)
4. (a) logb 5
(b) logb 15
(c) logb 3 0.4
(d) logb 4 60
5. (a) log3 b
(b) log3 (10b)
6. (a) log b2 5
(b) log
7. (a) log3b 2
(b) log3b 15
8. (a) log5b 1.2
(b) log5b 2.5
9. (a)
logb 5 log5 b
(b)
b
2
logb 6 log 6 b
10. (a) log2b 6 log2b (1/ 6) (b) log18 (1/ b)
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In exercises 11 and 12, suppose that log10 A a , log10 B b , and log10 C c
Express the following logarithms in terms of a, b, and/or c.
10 A
11. (a) log10 AB 2C 3
(b) log10 10 A
(c) log10 10ABC
(d) log10
BC
( AB)5
100 A2
(d)
log
10
C
B4 3 C
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In exercises 13 and 14, suppose that ln x t and ln y u .
Write each expression in terms of t and/or u.
12. (a) log10 A 2 log10
13. (a) ln(ex)
14. (a) ln e
ln x
1
A
(b) log10
(b) ln( xy) ln x 2
(b) e
ln (ln ( x y ))
A
10
(c) log10
(c) ln xy ln
ex
(c) ln
y
y
ln
ex
x
e
(d) ln e2 x y
(d)
ln x
ln
3
ln x 4
x
ln( xe 2 )
2
e
Answers:
1.
(a) A B
(b)
A B
(c) 3B
(d) -3B
(d) B A C
2. (a) A C
(b) 2 A 2C
(c)
3. (a) C B
(b) B C
(c) C 2B
d) C 4 A
4. (a) ½ C
(b) ½ B + ½ C
(c) ⅓ A - ⅓ C
(d)
5. (a)
1
B
(b)
6. (a)
C
2
(b) 2A
7. (a)
8. (a)
A
B 1
(b)
A B C
C 1
2 A 2C
1
1
1
A
B
C
2
4
4
A C 1
B
B C
B 1
(b)
9. (a) 1
(b) 1
10. (a) 0
(b)
C A
C 1
1
A 2B
11. (a) a 2b 3c
(b) 1 + ½ a
(c) ½ (1 + a + b + c)
(d) 1 + a – ½ b – ½ c
12. (a) – a
(b) a – 1
(c) 2 + 2a – 4b – ⅓ c
(d) 5a + 5b – c
13. (a) 1 + t
(b) u – t
(c)
14. (a) t
(b) t + u
(c) 2(1 + t – u)
3
t
2
1
u 1
2
(d) 2 + t + ½ u
(d) t