Logarithms Questions and Answers - Set 3

This set of Aptitude Questions and Answers (MCQs) focuses on “Logarithms – Set 3”.

1. What will be the sum of all the values of w which satisfy 2^log₇|w+9| = log₇2401?
a) -58
b) 40
c) -18
d) 18
View Answer

Answer: c
Explanation: Given,
2^log₇|w+9| = log₇2401
➩ 2^log₇|w+9| = log₇7⁴
➩ 2^log₇|w+9| = 4
➩ log₇|w + 9| = 2
➩ |w + 9| = 7² = 49
➩ w + 9 = 49 or w + 9 = -49
➩ w = 40 or -58
Thus, the required sum is 40 – 58 = -18.

2. Four natural numbers p, q, r, and s are less than 1000 such that log_pq = 7 / 5 and log_rs = 4 / 3. Now that s – q is given as 497 what will be the value of p + r?
a) 157
b) 165
c) 125
d) 139
View Answer

Answer: a
Explanation: Given,
p, q, r and s are natural numbers less than 1000.
➩ q = p^7/5
➩ only possible value of p would be 2⁵.
➩ Thus, q = 2⁷
➩ s = r^4/3 and s – q = 497 this gives s = 625.
➩ r = s^3/4 this gives r = 125
➩ p + r = 32 + 125 = 157

3. What will be the value of log 0.242424….?
a) -0.9956
b) -0.6154
c) -0.5881
d) -1.3412
View Answer

Answer: b
Explanation: Given,
log 0.242424 ……
➩ log 24 / 99
➩ log ((8 * 3) / (9 * 11))
➩ log ((2³ * 3) / (3² * 11))
➩ ((3log2 + log3) / (2log3 + log11))
➩ 3log2 + log3 – 2log3 – log11
➩ (3 * 0.3010) + 0.4771 – (2 * 0.4771) – (1.0413)
➩ 0.9030 – 0.4771 – 1.0413
➩ -0.6154

4. Which value of a would satisfy the expression log₉a + log₂₇√a = (log₄₉11) / (log₇3)?
a) 11^{3 / 4}
b) 7^{3 / 2}
c) 3^{9 / 11}
d) 5^{27 / 7}
View Answer

Answer: a
Explanation: LHS
log₉a + log₂₇√a
➩ \(\frac {1}{2}\) log₃a + 1 / 6 log₃a
➩ (2 / 3) log₃a
➩ (2 / 3) (log a / log3)
RHS
(log₄₉11) / (log₇3)
➩ ((1 / 2 log₇11) / log₇3)
➩ (1 / 2) ((log11 / log7) * (log7 / log3))
➩ (1 / 2) (log11 / log3)
On equating LHS and RHS
➩ (2 / 3) (log a / log3) = (1 / 2) (log11 / log3)
➩ (2 / 3) log a = (1 / 2) log 11
➩ a = (3 / 4) log 11
➩ a = 11^3/4

5. What will be the number of digits in the number 6²⁰?
a) 30
b) 32
c) 16
d) 15
View Answer

Answer: c
Explanation: Given,
6²⁰
⇒ log (6²⁰)
⇒ 20 log 6
⇒ 20 log (3 * 2)
⇒ 20 (log 3 + log 2)
⇒ 20 (0.4771 + 0.3010)
⇒ 20 * 0.7781
⇒ 15.562
Thus, the number of digits would be 15 + 1 = 16.

6. Which of the following values of z would not satisfy (log₅z)² + log_5z (5 / z) = 1?
a) 1
b) 5
c) 1 / 25
d) 10
View Answer

Answer: d
Explanation: Given,
(log₅z)² + log_5z (5 / z) = 1
➩ (log₅z)² + ((1 – log₅z) / (1 + log₅z)) = 1
Let log₅z = a.
➩ a² + ((1 – a) / (1 + a)) = 1
➩ a³ + a² + 1 – a = 1 + a
➩ a³ + a² – 2a = 0
Thus, a = 0, 1, -2.
log₅z = 0, 1, -2
➩ z = 1, 5, 1 / 25
Thus, any value of z apart from 1, 5 and 1 / 25 will not satisfy the given expression.

7. What will be the value of log_y441 if the value of log_y147 = A and log_y63 = B?
a) 2 (A + B) / 3
b) 2A + B / 6
c) A + B / 5
d) A² + B² / 3
View Answer

Answer: a
Explanation: Given,
log_y441
➩ log_y441 = log_y (49 * 9)
➩ log_y441 = log_y49 + log_y9
➩ log_y441 = 2 log_y7 + 2 log_y3
➩ let log_y7 = Q and log_y3 = R
➩ log_y441 = 2Q + 2R
Also, log_y147 = A and log_y63 = B
➩ R + 2Q = A and 2R + Q = B
➩ 2Q + 2R = 2 (A + B) / 3

8. What will be the value of log₂3 * log₃2 * log₃4 * log₄3 * log₅6 * log₆5?
a) 27
b) 3
c) 1
d) 9
View Answer

Answer: c
Explanation: Given,
log₂3 * log₃2 * log₃4 * log₄3 * log₅6 * log₆5
➩ (log3 / log2) * (log2 / log3) * (log4 / log3) * (log3 / log4) * (log6 / log5) * (log5 / log6)
On grouping gives,
➩ [(log3 / log2) * (log2 / log3)] * [(log4 / log3) * (log3 / log4)] * [(log6 / log5) * (log5 / log6)]
➩ 1 * 1 * 1
➩ 1

9. What will be the value of log 9 / 8 + log 3 / 4 – log 27 / 32?
a) 5
b) 10
c) 1
d) 0
View Answer

Answer: d
Explanation: Given,
log 9 / 8 + log 3 / 4 – log 27 / 32
➩ log 3² / 2³ + log 3 / 2² – log 3³ / 2⁵
➩ 2log 3 / 3 log2 + log 3 / 2 log 2 – 3 log 3 / 5 log 2
➩ log 1
➩ 0

10. Which of the following values of t would satisfy the expression log₁₀(25 – t) – log₂4 = log₁₀t – 2 log₁₀5?
a) 5
b) 10
c) 8
d) 12
View Answer

Answer: a
Explanation: Given,
log₁₀(25 – t) – log₂4 = log₁₀t – 2 log₁₀5
LHS
➩ log₁₀(25 – t) – log₂4
➩ log₁₀(25 – t) – log₂2²
➩ log₁₀(25 – t) – 2
➩ log₁₀(25 – t) – log₁₀ 100
➩ log₁₀((25 – t) / 100)
RHS
➩ log₁₀t – 2 log₁₀5
➩ log₁₀t – log₁₀5²
➩ log₁₀t – log₁₀25
➩ log₁₀(t / 25)
On equating RHS and LHS gives,
➩ t / 25 = 25 – t / 100
➩ t = 25 – t / 4
➩ 4t = 25 – t
➩ 5t = 25
➩ t = 5

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