Logarithms Questions and Answers - Expanding and Condensing Logarithm - Set 2

This set of Aptitude Questions and Answers (MCQs) focuses on “Expanding and Condensing Logarithm – Set 2”.

1. Which of the following options is the correct simplified form of the expression log (75 / 16) – 2log (5 / 9) + log (32 / 343)?
a) log (2 / 3)
b) log (5 / 3)
c) log 2
d) 3 / 19
View Answer

Answer: c
Explanation: Given,
➩ log (75 / 16) – 2log (5 / 9) + log (32 / 343)
➩ log (25 * 3 / 4 * 4) – log (25 / 81) + log ((16 * 2) / (81 * 3))
➩ log (25 * 3) – log (4 * 4) – log (25) + log (81) + log (16 * 2) – log (81 * 3)
➩ log 25 + log 3 – log 16 – log 25 + log 81 + log 16 + log 2 – log 81 – log 3
➩ on simplifying gives log 2.

2. What is the value of log₁₀1.5?
a) 0.1751
b) 0.2468
c) 0.8423
d) 0.3752
View Answer

Answer: a
Explanation: Given,
log₁₀1.5
➩ log₁₀ (15 / 10)
➩ log 15 – log 10
➩ log (5 * 3) – log 10
➩ log 5 + log 3 – log 10
➩ 0.698 + 0.4771 – 1
➩ 1.1751 – 1
➩ 0.1751

3. Which of the following is the correct solution of the expression log (x² / yz) + log (y² / xz) + log (z² / xy)?
a) x²y²z²
b) 0
c) 1
d) xyz
View Answer

Answer: b
Explanation: Given,
log (x² / yz) + log (y² / xz) + log (z² / xy)
➩ log ((x² * y² * z²) / (yz * xz * xy))
➩ log ((x² * y² * z²) / (x² * y² * z²))
➩ log 1
➩ 0

4. Which of the following is a correct alternate representation of the expression log_c z + log_c (1 + z) = 0?
a) z² + z – 1 = 0
b) z² + 2z + 2 = 0
c) z³ + z² + z = 0
d) z + 5 = 0
View Answer

Answer: a
Explanation: Given,
log_c z + log_c (1 + z) = 0
➩ log_c z (1 + z) = 0
➩ log_c z (1 + z) = log_c1
➩ z (1 + z) = 1
➩ z + z² = 1
➩ z² + z – 1 = 0

5. Which values of m would satisfy the equation 2log₂m = 1 + log₂((5m / 2) – 3)?
a) 4, 1
b) 5, 3
c) 3, 2
d) 2, 9
View Answer

Answer: c
Explanation: Given,
2log₂m = 1 + log₂((5m / 2) – 3)
➩ log₂m² = log₂2 + log₂((5m / 2) – 3)
➩ m² = 2 ((5m / 2) – 3)
➩ m² = 5m – 6
➩ m² – 5m + 6 = 0
➩ m² – 3m – 2m + 6 = 0
➩ m (m – 3) – 2 (m – 3) = 0
➩ (m – 3) (m – 2) = 0
➩ m = 3, 2

6. Which of the following options represent the right value of variable z if, (10000)^z = 4?
a) 0.5692
b) 0.2558
c) 0.3010
d) 0.1505
View Answer

Answer: d
Explanation: Given,
(10000)^z = 4
➩ z log 10⁴ = log 4
➩ 4z = log 4
➩ z = log 4 / 4
➩ z = log 2² / 4
➩ z = 2 log 2 / 4
➩ z = log 2 / 2
➩ z = 0.3010 / 2
➩ z = 0.1505

7. Which of the following values of a satisfies the expression 5^5-a = 2^-(5-a)?
a) 5
b) 2
c) 4
d) Can’t be determined
View Answer

Answer: a
Explanation: Given,
5^5-a = 2^-(5-a)
➩ (5 – a) log 5 = – (5 – a) log 2
➩ (5 – a) log 5 + (5 – a) log 2 = 0
➩ (5 – a) (log 5 + log 2) = 0
➩ (5 – a) (log 10 / 2 + log 2) = 0
➩ (5 – a) (log 10 – log 2 + log 2) = 0
➩ (5 – a) (log 10) = 0
➩ 5 – a = 0
➩ a = 5

8. What will be the value of log_0.12532?
a) -3.54
b) 3.54
c) -1.67
d) 1.67
View Answer

Answer: c
Explanation: Given,
log_0.12532
➩ 0.125 can be written as 2^-3.
➩ 32 can be written as 2⁵.
Thus, the expression can be written as:
➩ log₂^-3 2⁵
➩ (5 / (- 3)) log₂2
➩ (5 / (- 3)) * 1
➩ (5 / (- 3))
➩ -1.67

9. What will be the value of log₁₀10 + log₁₀1000 + log₁₀100000 + log₁₀10000000?
a) 16 log₁₀100
b) 16
c) log₁₆10
d) 132
View Answer

Answer: b
Explanation: Given,
log₁₀10 + log₁₀1000 + log₁₀100000 + log₁₀10000000
➩ log₁₀10¹ + log₁₀10³ + log₁₀10⁵ + log₁₀10⁷
➩ 1 log₁₀10 + 3 log₁₀10 + 5 log₁₀10 + 7 log₁₀10
➩ 1 + 3 + 5 + 7
➩ 16

10. What will be the value of b if log₁₀a + log₁₀b = c?
a) 10^c / a
b) a / c
c) a²c² / 2
d) 10^a / c
View Answer

Answer: a
Explanation: Given,
log₁₀a + log₁₀b = c
➩ log₁₀ (ab) = c
➩ ab = 10^c
➩ b = 10^c / a

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