Logarithms are a fundamental concept in mathematics, particularly in algebra and calculus. A logarithm is the power to which a number, called the base, must be raised to obtain another number. Understanding logarithms is essential for solving exponential equations, growth decay problems, and in various scientific applications.
1. If log10 2 = 0.3010, find the value of log10 5.
a) 0.6989
b) 0.6990
c) 0.7000
d) 0.7010
Answer:
a) 0.6989
Explanation:
Using the property log10(a/b) = log10 a - log10 b, log10 5 = log10(10/2) = log10 10 - log10 2 = 1 - 0.3010 = 0.6989.
2. Find the value of log2 8.
a) 2
b) 3
c) 4
d) 5
Answer:
b) 3
Explanation:
log2 8 = x implies 2^x = 8. Since 2^3 = 8, x = 3.
3. If log3 (9x) = 4, find x.
a) 27
b) 36
c) 45
d) 81
Answer:
d) 81
Explanation:
log3 (9x) = 4 implies 3^4 = 9x.
81 = 9x, thus x = 81 / 9 = 9.
4. Calculate log5 125.
a) 2
b) 3
c) 4
d) 5
Answer:
b) 3
Explanation:
log5 125 = x implies 5^x = 125. Since 5^3 = 125, x = 3.
5. If log x = 0.3010, find the value of x.
a) 2
b) 5
c) 10
d) 20
Answer:
a) 2
Explanation:
log x = 0.3010 implies x = 10^0.3010. Since log10 2 ≈ 0.3010, x ≈ 2.
6. Evaluate log10 1000.
a) 1
b) 2
c) 3
d) 4
Answer:
c) 3
Explanation:
log10 1000 = log10(10^3) = 3.
7. Find the value of log4 64.
a) 2
b) 3
c) 4
d) 5
Answer:
b) 3
Explanation:
log4 64 = x implies 4^x = 64. Since 4^3 = 64, x = 3.
8. If log2 x = 5, find x.
a) 16
b) 32
c) 64
d) 128
Answer:
b) 32
Explanation:
log2 x = 5 implies x = 2^5 = 32.
9. What is the value of log10(1/100)?
a) -1
b) -2
c) -3
d) -4
Answer:
b) -2
Explanation:
log10(1/100) = log10(10^-2) = -2.
10. Find log7(1/49).
a) -1
b) -2
c) -3
d) -4
Answer:
b) -2
Explanation:
log7(1/49) = log7(7^-2) = -2.
11. If log3 x – log3 27 = 2, find x.
a) 81
b) 243
c) 729
d) 2187
Answer:
b) 243
Explanation:
log3 x - log3 27 = 2 implies log3 x = log3 27 + 2.
x = 3^(3+2) = 3^5 = 243.
12. Calculate log2(1/32).
a) -3
b) -4
c) -5
d) -6
Answer:
c) -5
Explanation:
log2(1/32) = log2(2^-5) = -5.
13. What is the value of log(0.0001)?
a) -2
b) -3
c) -4
d) -5
Answer:
c) -4
Explanation:
log(0.0001) = log(10^-4) = -4.
14. Find log5 25.
a) 1
b) 1.5
c) 2
d) 2.5
Answer:
c) 2
Explanation:
log5 25 = log5(5^2) = 2.
15. If log(2x) = 3, find x.
a) 500
b) 1000
c) 1500
d) 2000
Answer:
b) 1000
Explanation:
log(2x) = 3 implies 2x = 10^3.
x = 1000 / 2 = 500.
16. Evaluate log2 16.
a) 2
b) 3
c) 4
d) 5
Answer:
c) 4
Explanation:
log2 16 = log2(2^4) = 4.
17. If log x + log y = log 50 and x*y = 50, find x and y.
a) x = 5, y = 10
b) x = 10, y = 5
c) x = 25, y = 2
d) x = 2, y = 25
Answer:
b) x = 10, y = 5
Explanation:
log x + log y = log 50 implies x*y = 50. Possible pairs (x, y) are (1, 50) and (10, 5). The pair (10, 5) satisfies both conditions.
18. Simplify: log(100/10).
a) 1
b) 2
c) 0
d) 10
Answer:
a) 1
Explanation:
log(100/10) = log(10) = 1.
19. If log4(x^2) = 4, find x.
a) 4
b) 8
c) 16
d) 32
Answer:
b) 8
Explanation:
log4(x^2) = 4 implies x^2 = 4^4.
x^2 = 256, thus x = √256 = 16.
20. Calculate log2 1.
a) 0
b) 1
c) 2
d) 3
Answer:
a) 0
Explanation:
log2 1 = 0, as 2^0 = 1.