Logarithms Questions and Answers - Change of Base

This set of Aptitude Questions and Answers (MCQs) focuses on “Change of Base”.

1. Which value of x would satisfy 2^{3x + 3} = 8^{2x + 2}?
a) 3 / 7
b) -2 / 3
c) -1
d) 3 / 19
View Answer

Answer: c
Explanation: Given,
2^{3x + 3} = 8^{2x + 2}
ln 2^{3x + 3} = ln 2^{3(2x + 2)}
3x + 3 = 3(2x + 2)
3x + 3 = 6x + 6
3x = -3
x = -1

2. Which of the following options gives the correct value of log₇(1 / 2401)?
a) -4
b) -7
c) -8
d) -3
View Answer

Answer: a
Explanation: Let,
log₇(1 / 2401) = y
➩ 7^y = 1 / 2401
➩ 7^y = 1 / 7⁴
➩ 7^y = 7^-4
➩ Y = -4

3. Which of the given options is the correct simplified value of log₄3 * log₂₇16?
a) 3 / 4
b) 2 / 3
c) 2 / 5
d) 5 / 4
View Answer

Answer: b
Explanation: Given,
log₄3 * log₂₇16
➩ (log 3 / log 4) * (log 16 / log 27)
➩ (log 3 / log 4) * (log 4² / log 3³)
➩ (log 3 / log 4) * (2(log 4) / 3(log 3))
➩ 2 / 3

4. If log_√243a = 8 / 5, then what would be the true value of a from the options given below?
a) 81
b) 78
c) 64
d) 47
View Answer

Answer: a
Explanation: Given,
log_√243a = 8 / 5
➩ (√243)^(8/5) = a
➩ (3^(5/2))^(8/5) = a
➩ (3^(5/2)(8/5)) = a
➩ 3⁴ = a
➩ 81 = a

5. Which value of p would satisfy [log 2 + log (4p + 1) = log (p + 2) + 1]?
a) 10
b) -12
c) -9
d) 14
View Answer

Answer: c
Explanation: Given,
[log 2 + log (4p + 1) = log (p + 2) + 1]
➩ log 2 + log (4p + 1) = log (p + 2) + log 10
➩ log 2 (4p + 1) = log 10 (p + 2)
➩ 2 (4p + 1) = 10 (p + 2)
➩ 8p + 2 = 10p + 20
➩ 2p = -18
➩ p = -9

6.Which of the following options is the correct depiction of log (8 * (147)^1/3), if log2 = a, log3 = b and log₇ = c?
a) 3a + (1 / 3) c + (3 / 4) b
b) 2a + (3 / 4) c + (1 / 6) b
c) a + 2b + 3c
d) 3a + (2 / 3) c + (1 / 3) b
View Answer

Answer: d
Explanation: Given,
log (8 * (147)^1/3)
➩ log 8 + log (147)^1/3
➩ log 2³ + log (7² * 3)^1/3
➩ 3log 2 + (2 / 3) log 7 + (1 / 3) log 3
As, log2 = a, log3 = b and log₇ = c
➩ 3a + (2 / 3) c + (1 / 3) b

7. Which value of z would satisfy the expression log₂₄₀₁z = – (1 / 4)?
a) 1 / 7
b) 35000
c) 45000
d) 50000
View Answer

Answer: a
Explanation: Given,
log₂₄₀₁z = -(1 / 4)
➩ z = (2401)^-1/4
➩ z = (7⁴)^(-1/4)
➩ z = 7 – 1
➩ z = 1 / 7

8. Which value of z would satisfy the expression log₂₇ (3log₂ (1 + log₃ (1 + 2log₂z)) = 1 / 3?
a) 6
b) 3
c) 2
d) 5
View Answer

Answer: c
Explanation: Given,
Log₂₇ (3log₂ (1 + log₃ (1 + 2log₂z)) = 1 / 3
➩ 3log₂ (1 + log₃ (1 + 2log₂z)) = 27^1/3 = 3
➩ log₂ (1 + log₃ (1 + 2log₂z)) = 1
➩ 1 + log₃ (1 + 2log₂z) = 2
➩ log₃ (1 + 2log₂z) = 1
➩ 1 + 2log₂z = 3
➩ 2log₂z = 2
➩ log₂z = 1
➩ z = 2

9. Given the value of log₅z = p and log₂₀z = q, then what should be the value of log_z10?
a) ((ab) / a + b)
b) ((a + b) / 2ab)
c) a² / b³
d) ((a – b) / (b²a))
View Answer

Answer: b
Explanation: Given,
log₅z = p and log₂₀z = q
➩ log_z5 = 1 / p, log_z20 = 1 / q
➩ log_z (100 / 5) = 1 / b
➩ log_z100 – log_z5 = 1 / b
➩ 2log_z10 – 1 / a = 1 / b
➩ log_z10 = \(\frac {1}{2}\) (1 / a + 1 / b)
➩ log_z10 = ((a + b) / 2ab)

10. Provided log₄16 + log₄4 = z, then what is the value of z?
a) 3
b) 4
c) 8
d) 16
View Answer

Answer: a
Explanation: Given,
log₄16 + log₄4 = z
➩ log₄ (16 * 4) = z
➩ z = log₄ (64)
➩ z = log₄ (4³)
➩ z = 3 log₄4
➩ z = 3

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