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

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 log_{7}(1 / 2401)?

a) -4

b) -7

c) -8

d) -3

View Answer

Explanation: Let,

log

_{7}(1 / 2401) = y

➩ 7

^{y}= 1 / 2401

➩ 7

^{y}= 1 / 7

^{4}

➩ 7

^{y}= 7

^{-4}

➩ Y = -4

3. Which of the given options is the correct simplified value of log_{4}3 * log_{27}16?

a) 3 / 4

b) 2 / 3

c) 2 / 5

d) 5 / 4

View Answer

Explanation: Given,

log

_{4}3 * log

_{27}16

➩ (log 3 / log 4) * (log 16 / log 27)

➩ (log 3 / log 4) * (log 4

^{2}/ log 3

^{3})

➩ (log 3 / log 4) * (2(log 4) / 3(log 3))

➩ 2 / 3

4. If log_{√243}a = 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

Explanation: Given,

log

_{√243}a = 8 / 5

➩ (√243)

^{(8/5)}= a

➩ (3

^{(5/2)})

^{(8/5)}= a

➩ (3

^{(5/2)(8/5)}) = a

➩ 3

^{4}= 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

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 log_{7} = 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

Explanation: Given,

log (8 * (147)

^{1/3})

➩ log 8 + log (147)

^{1/3}

➩ log 2

^{3}+ log (7

^{2}* 3)

^{1/3}

➩ 3log 2 + (2 / 3) log 7 + (1 / 3) log 3

As, log2 = a, log3 = b and log

_{7}= c

➩ 3a + (2 / 3) c + (1 / 3) b

7. Which value of z would satisfy the expression log_{2401}z = – (1 / 4)?

a) 1 / 7

b) 35000

c) 45000

d) 50000

View Answer

Explanation: Given,

log

_{2401}z = -(1 / 4)

➩ z = (2401)

^{-1/4}

➩ z = (7

^{4})

^{(-1/4)}

➩ z = 7 – 1

➩ z = 1 / 7

8. Which value of z would satisfy the expression log_{27} (3log_{2} (1 + log_{3} (1 + 2log_{2}z)) = 1 / 3?

a) 6

b) 3

c) 2

d) 5

View Answer

Explanation: Given,

Log

_{27}(3log

_{2}(1 + log

_{3}(1 + 2log

_{2}z)) = 1 / 3

➩ 3log

_{2}(1 + log

_{3}(1 + 2log

_{2}z)) = 27

^{1/3}= 3

➩ log

_{2}(1 + log

_{3}(1 + 2log

_{2}z)) = 1

➩ 1 + log

_{3}(1 + 2log

_{2}z) = 2

➩ log

_{3}(1 + 2log

_{2}z) = 1

➩ 1 + 2log

_{2}z = 3

➩ 2log

_{2}z = 2

➩ log

_{2}z = 1

➩ z = 2

9. Given the value of log_{5}z = p and log_{20}z = q, then what should be the value of log_{z}10?

a) ((ab) / a + b)

b) ((a + b) / 2ab)

c) a^{2} / b^{3}

d) ((a – b) / (b^{2}a))

View Answer

Explanation: Given,

log

_{5}z = p and log

_{20}z = q

➩ log

_{z}5 = 1 / p, log

_{z}20 = 1 / q

➩ log

_{z}(100 / 5) = 1 / b

➩ log

_{z}100 – log

_{z}5 = 1 / b

➩ 2log

_{z}10 – 1 / a = 1 / b

➩ log

_{z}10 = \(\frac {1}{2}\) (1 / a + 1 / b)

➩ log

_{z}10 = ((a + b) / 2ab)

10. Provided log_{4}16 + log_{4}4 = z, then what is the value of z?

a) 3

b) 4

c) 8

d) 16

View Answer

Explanation: Given,

log

_{4}16 + log

_{4}4 = z

➩ log

_{4}(16 * 4) = z

➩ z = log

_{4}(64)

➩ z = log

_{4}(4

^{3})

➩ z = 3 log

_{4}4

➩ z = 3

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