Logarithms Questions and Answers - Expanding and Condensing Logarithm

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

1. What would be the value of x if log (x – 1) + log (x + 1) = log 100?
a) (99)^1/2
b) -(101)^1/2
c) (101)^1/2
d) (99)^-1/2
View Answer

Answer: c
Explanation: Given,
log (x – 1) + log (x + 1) = log 100
log (x – 1) + log (x + 1) = log 10²
log (x – 1) + log (x + 1) = 2log 10
log ((x – 1) (x + 1)) = 2
(x – 1) (x + 1) = 100
x² – 1 = 100
x² = 101
x = +(101)^1/2 or -(101)^1/2
Since, log of negative values is not defined therefore, x = (101)^1/2.

2. What would be the value of z, if log (z – 30) = 1000?
a) 33
b) 27
c) 36
d) 24
View Answer

Answer: a
Explanation: Given,
➩ log (z – 30) = 1000
➩ log (z – 30) = log 10³
➩ log (z – 30) = 3log 10
➩ z – 30 = 3
➩ z = 3 + 30 = 33

3. From the given values of ‘x’ which value would satisfy log (x + 3) + log (x – 3) = 4log2?
a) 4
b) 5
c) 9
d) 16
View Answer

Answer: b
Explanation: Given,
log (x + 3) + log (x – 3) = 4log2
➩ log ((x + 3) (x – 3)) = log2⁴
➩ log (x² – 9) = log16
➩ x² – 9 = 16
➩ x² = 16 + 9
➩ x² = 25
➩ x = +5, -5

4. Given log_xa = b, then x^b-1 can be expressed as which of the following?
a) a / b
b) b / a
c) b² / a
d) a³ / 3b
View Answer

Answer: a
Explanation: Given,
log_xa = b
➩ x^b = a
➩ x^b / x = a / b
➩ x^b-1 = a / b

5. Given log₈a = 25 and log₂b = 5, then which of the following options is correct?
a) a²b = 2 / 3
b) ab = 1
c) a = b¹⁵
d) b = a¹⁵
View Answer

Answer: c
Explanation: Given,
log₈a = 25 and log₂b = 5
➩ a = 8²⁵ and b = 2⁵
➩ a = (2³)²⁵ and b = 2⁵
➩ a = 2⁷⁵ and b = 2⁵
➩ a = (2⁵)¹⁵ and b = 2⁵
➩ a = b¹⁵

6. Which of the following is the correct simplified form of the equation (4 log₁₀10000) / (2 log₁₀100)?
a) 2
b) 16
c) 8
d) 4
View Answer

Answer: d
Explanation: Given,
(4 log₁₀10000) / (2 log₁₀100)
➩ (4 log₁₀10⁴) / (2 log₁₀10²)
➩ (4 * 4 log₁₀10) / (2 * 2 log₁₀10)
➩ 16 / 4
➩ 4

7. What would be the value of log₅12, if log₁₀2 = x and log₁₀3 = y?
a) (2x + y) / (1 – x)
b) (2y + x) / (1 – y)
c) xy / (y + 1)
d) x²y² / (x + 1)
View Answer

Answer: a
Explanation: Given,
log₁₀2 = x and log₁₀3 = y
➩ log₅12 = log₅ (3 * 4)
= log₅3 + log₅4
= log₅3 + log₅2²
= log₅3 + 2log₅2
= ((log₁₀3) / (log₁₀5)) + ((2log₁₀2) / (log₁₀5))
= ((log₁₀3) / (log₁₀10 – log₁₀2)) + ((2log₁₀2) / (log₁₀10 – log₁₀2))
= (y / 1 – x) + (2x / 1 – x)
= (2x + y) / (1 – x)

8. What could be the value of z, if log₅ (z² + z) – log₅ (z + 1) = 2?
a) 30
b) 20
c) 25
d) 4
View Answer

Answer: c
Explanation: Given,
log₅ (z² + z) – log₅ (z + 1) = 2
➩ log₅ ((z² + z) / (z + 1)) = 2
➩ log₅ (z(z + 1) / (z + 1)) = 2
➩ log₅z = 2
➩ z = 5²
➩ z = 25

9. If 1 / 2 (log p + log q) = log ((p + q) / 2), then which relationship between a and b would be correct?
a) p = q
b) p² = q
c) p = q / 2
d) q = p / 3
View Answer

Answer: a
Explanation: Given,
1 / 2 (log p + log q) = log ((p + q) / 2)
➩ 1 / 2 log (pq) = log ((p + q) / 2)
➩ log (pq)^1/2 = log ((p + q) / 2)
➩ (pq)^1/2 = (p + q) / 2
➩ pq = ((p + q) / 2)²
➩ 4pq = p² + q² + 2pq
➩ P² + q² + – 2pq = 0
➩ (p – q)² = 0
➩ p – q = 0
➩ p = q

10. What will be the value of log₁₀ (p^xq^y), given log₁₀p = x and log₁₀q = y?
a) x² + y²
b) xy²
c) 3x + y³
d) 3xy / 4
View Answer

Answer: a
Explanation: Given,
log₁₀p = x and log₁₀q = y
log₁₀ (p^xq^y) = log₁₀p^x + log₁₀q^y
= xlog₁₀p + ylog₁₀q
= x² + y²

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