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

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 log_{10}1.5?

a) 0.1751

b) 0.2468

c) 0.8423

d) 0.3752

View Answer

Explanation: Given,

log

_{10}1.5

➩ log

_{10}(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 (x^{2} / yz) + log (y^{2} / xz) + log (z^{2} / xy)?

a) x^{2}y^{2}z^{2}

b) 0

c) 1

d) xyz

View Answer

Explanation: Given,

log (x

^{2}/ yz) + log (y

^{2}/ xz) + log (z

^{2}/ xy)

➩ log ((x

^{2}* y

^{2}* z

^{2}) / (yz * xz * xy))

➩ log ((x

^{2}* y

^{2}* z

^{2}) / (x

^{2}* y

^{2}* z

^{2}))

➩ 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) z^{2} + z – 1 = 0

b) z^{2} + 2z + 2 = 0

c) z^{3} + z^{2} + z = 0

d) z + 5 = 0

View Answer

Explanation: Given,

log

_{c}z + log

_{c}(1 + z) = 0

➩ log

_{c}z (1 + z) = 0

➩ log

_{c}z (1 + z) = log

_{c}1

➩ z (1 + z) = 1

➩ z + z

^{2}= 1

➩ z

^{2}+ z – 1 = 0

5. Which values of m would satisfy the equation 2log_{2}m = 1 + log_{2}((5m / 2) – 3)?

a) 4, 1

b) 5, 3

c) 3, 2

d) 2, 9

View Answer

Explanation: Given,

2log

_{2}m = 1 + log

_{2}((5m / 2) – 3)

➩ log

_{2}m

^{2}= log

_{2}2 + log

_{2}((5m / 2) – 3)

➩ m

^{2}= 2 ((5m / 2) – 3)

➩ m

^{2}= 5m – 6

➩ m

^{2}– 5m + 6 = 0

➩ m

^{2}– 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

Explanation: Given,

(10000)

^{z}= 4

➩ z log 10

^{4}= log 4

➩ 4z = log 4

➩ z = log 4 / 4

➩ z = log 2

^{2}/ 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

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_{0.125}32?

a) -3.54

b) 3.54

c) -1.67

d) 1.67

View Answer

Explanation: Given,

log

_{0.125}32

➩ 0.125 can be written as 2

^{-3}.

➩ 32 can be written as 2

^{5}.

Thus, the expression can be written as:

➩ log

_{2}

^{-3}2

^{5}

➩ (5 / (- 3)) log

_{2}2

➩ (5 / (- 3)) * 1

➩ (5 / (- 3))

➩ -1.67

9. What will be the value of log_{10}10 + log_{10}1000 + log_{10}100000 + log_{10}10000000?

a) 16 log_{10}100

b) 16

c) log_{16}10

d) 132

View Answer

Explanation: Given,

log

_{10}10 + log

_{10}1000 + log

_{10}100000 + log

_{10}10000000

➩ log

_{10}10

^{1}+ log

_{10}10

^{3}+ log

_{10}10

^{5}+ log

_{10}10

^{7}

➩ 1 log

_{10}10 + 3 log

_{10}10 + 5 log

_{10}10 + 7 log

_{10}10

➩ 1 + 3 + 5 + 7

➩ 16

10. What will be the value of b if log_{10}a + log_{10}b = c?

a) 10^{c} / a

b) a / c

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

d) 10^{a} / c

View Answer

Explanation: Given,

log

_{10}a + log

_{10}b = c

➩ log

_{10}(ab) = c

➩ ab = 10

^{c}

➩ b = 10

^{c}/ a

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