Logarithms Questions and Answers - Properties of Logarithm

This set of Aptitude Questions and Answers (MCQs) focuses on “Properties of Logarithm”.

1. What will be the value of log₃8⁵?
a) 9.863
b) 7.527
c) 9.463
d) 8.321
View Answer

Answer: c
Explanation: Given,
log₃8⁵
log 8⁵ / log3
5 log 2³ / log3
5 * 3 log 2 / log3
15 * 0.3010 / 0.4771
4.515 / 0.4771
9.463

2. What will be the value of integer z if (log₂z)² – log₂z⁴ – 32 = 0?
a) 256
b) 64
c) 16
d) 1024
View Answer

Answer: a
Explanation: Given,
(log₂z)² – log₂z⁴ – 32 = 0
➩ (log₂z)² – 4log₂z – 32 = 0
Let log₂z = a
➩ a² – 4a – 32 = 0
➩ a² – 8a + 4a – 32 = 0
➩ a(a – 8) + 4(a – 8) = 0
➩ (a – 8) (a + 4) = 0
➩ a = 8, – 4
➩ log₂z = 8 or log₂z = – 4
➩ z = 2⁸ = 256 or z = 2^-4 = 1 / 16
Since z is an integer therefore z = 256.

3. If (1 / 4) log₂a + 4log₂b = 2 + log₆₄8, then which of the following is true?
a) a¹⁶ = 64 / b³
b) b¹⁶ = 64 / a
c) a¹⁶ = 64 / b
d) b¹⁶ = 64 / a⁴
View Answer

Answer: b
Explanation: Given,
(1 / 4) log₂a + 4log₂b = 2 + log₆₄8
➩ (1 / 4) log₂a + 4log₂b = 2 – 1 / log₈64
➩ log₂ (a^1/4 * b⁴) = 2 – \(\frac {1}{2}\) = 3 / 2
➩ a^1/4 * b⁴ = 2^3/2
➩ ab¹⁶ = 64
➩ b¹⁶ = 64 / a

4. What will be the value log (8)^1/3 / log (4)²?
a) 1 / 4
b) 1 / 8
c) 1 / 3
d) 1 / 6
View Answer

Answer: a
Explanation: Given,
log (8)^1/3 / log (4)²
➩ log (2³)^1/3 / log (2²)²
➩ log (2)^3*1/3 / log (2)⁴
➩ log (2)^3/3 / log (2)⁴
➩ log (2) / log (2)⁴
➩ log (2) / 4 log (2)
➩ (1 / 4) (log (2) / log (2))
➩ 1 / 4

5. Which of the following statements is not correct?
a) log₁₀1 = 0
b) log₁₀10 = 1
c) log (5 + 7 + 9) = log5 + log7 + log9
d) log (3 + 4) = log (3 * 4)
View Answer

Answer: d
Explanation: Considering option d,
➩ log (3 + 4) = log 7
➩ log (3 * 4) = log 12
As, log 7 is not equal to log 12 therefore log (3 + 4) = log (3 * 4) is incorrect.

6. What will be the value of log₁₀ (1 / 500) if log₁₀5 = z?
a) z – 4
b) z³ / 5
c) z – 3 / z
d) –(z + 2)
View Answer

Answer: d
Explanation: Given,
log₁₀ (1 / 500)
➩ log₁₀1 – log₁₀500
➩ -log₁₀ (5 * 100)
➩ -(log₁₀5 + log₁₀100)
➩ -(log₁₀5 + log₁₀10²)
➩ -(log₁₀5 + 2 log₁₀10)
➩ -(z + 2)

7. Which value of a would satisfy log5 + log (a + 5) = log (5a + 10) + 1?
a) –1.66
b) 2.76
c) –3.45
d) –1.92
View Answer

Answer: a
Explanation: Given,
log5 + log (a + 5) = log (5a + 10) + 1
➩ log (5(a + 5)) = log (5a + 10) + log₁₀
➩ log (5(a + 5)) = log (10(5a + 10))
➩ 5(a + 5) = 10(5a + 10)
➩ 5a + 25 = 50a + 100
➩ -45 a = 75
➩ a = -(75 / 45)
➩ a = -1.66

8. What will be the number of digits in 3⁴⁵?
a) 28
b) 21
c) 22
d) 23
View Answer

Answer: c
Explanation: log (3⁴⁵)
➩ 45 * log 3
➩ 45 * 0.4771
➩ 21.4695
Thus, the characteristic is 21. Therefore, there will be 21 + 1 = 22 digits in 3⁴⁵.

9. What will be the value of p if log₅16, log₅(3^p – 4), log₅(3^p + 97 / 16) are in arithmetic progression?
a) 9
b) 4
c) 2
d) 3
View Answer

Answer: d
Explanation: Given,
log₅16, log₅(3^p – 4), log₅(3^p + 97 / 16) are in arithmetic progression.
➩ 2 log₅ (3^p – 4) = log₅16 + log₅(3^p + 97 / 16)
➩ log₅ (3^p – 4)² = log₅ [16(3^p + 97 / 16)]
➩ (3^p – 4)² = 16(3^p + 97 / 16)
Let 3^p = q
➩ (q – 4)² = 16(q + 97 / 16)
➩ q² – 8q + 16 = 16q + 97
➩ q² – 24q – 81 = 0
➩ q² – 27q + 3q – 81 = 0
➩ q (q – 27) + 3(q – 27) = 0
➩ (q + 3) (q – 27) = 0
➩ q = -3, 27
As 3^p is a positive number therefore, rejecting – 3.
Thus, 3^p = 27
➩ 3^p = 3³ ⟹ p = 3

10. What will be the value of y if log₂(log₂(log₃(log₂y))) = 1?
a) 2⁸¹
b) 3³²
c) 8³
d) 3⁶
View Answer

Answer: a
Explanation: Given,
log₂(log₂(log₃(log₂y))) = 1
➩ log₂(log₃(log₂y)) = 2¹ = 2
➩ log₃(log₂y) = 2² = 4
➩ log₂y = 3⁴ = 81
➩ y = 2⁸¹

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