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

Explanation: Given,

log (x – 1) + log (x + 1) = log 100

log (x – 1) + log (x + 1) = log 10

^{2}

log (x – 1) + log (x + 1) = 2log 10

log ((x – 1) (x + 1)) = 2

(x – 1) (x + 1) = 100

x

^{2}– 1 = 100

x

^{2}= 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

Explanation: Given,

➩ log (z – 30) = 1000

➩ log (z – 30) = log 10

^{3}

➩ 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

Explanation: Given,

log (x + 3) + log (x – 3) = 4log2

➩ log ((x + 3) (x – 3)) = log2

^{4}

➩ log (x

^{2}– 9) = log16

➩ x

^{2}– 9 = 16

➩ x

^{2}= 16 + 9

➩ x

^{2}= 25

➩ x = +5, -5

4. Given log_{x}a = b, then x^{b-1} can be expressed as which of the following?

a) a / b

b) b / a

c) b^{2} / a

d) a^{3} / 3b

View Answer

Explanation: Given,

log

_{x}a = b

➩ x

^{b}= a

➩ x

^{b}/ x = a / b

➩ x

^{b-1}= a / b

5. Given log_{8}a = 25 and log_{2}b = 5, then which of the following options is correct?

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

b) ab = 1

c) a = b^{15}

d) b = a^{15}

View Answer

Explanation: Given,

log

_{8}a = 25 and log

_{2}b = 5

➩ a = 8

^{25}and b = 2

^{5}

➩ a = (2

^{3})

^{25}and b = 2

^{5}

➩ a = 2

^{75}and b = 2

^{5}

➩ a = (2

^{5})

^{15}and b = 2

^{5}

➩ a = b

^{15}

6. Which of the following is the correct simplified form of the equation (4 log_{10}10000) / (2 log_{10}100)?

a) 2

b) 16

c) 8

d) 4

View Answer

Explanation: Given,

(4 log

_{10}10000) / (2 log

_{10}100)

➩ (4 log

_{10}10

^{4}) / (2 log

_{10}10

^{2})

➩ (4 * 4 log

_{10}10) / (2 * 2 log

_{10}10)

➩ 16 / 4

➩ 4

7. What would be the value of log_{5}12, if log_{10}2 = x and log_{10}3 = y?

a) (2x + y) / (1 – x)

b) (2y + x) / (1 – y)

c) xy / (y + 1)

d) x^{2}y^{2} / (x + 1)

View Answer

Explanation: Given,

log

_{10}2 = x and log

_{10}3 = y

➩ log

_{5}12 = log

_{5}(3 * 4)

= log

_{5}3 + log

_{5}4

= log

_{5}3 + log

_{5}2

^{2}

= log

_{5}3 + 2log

_{5}2

= ((log

_{10}3) / (log

_{10}5)) + ((2log

_{10}2) / (log

_{10}5))

= ((log

_{10}3) / (log

_{10}10 – log

_{10}2)) + ((2log

_{10}2) / (log

_{10}10 – log

_{10}2))

= (y / 1 – x) + (2x / 1 – x)

= (2x + y) / (1 – x)

8. What could be the value of z, if log_{5} (z^{2} + z) – log_{5} (z + 1) = 2?

a) 30

b) 20

c) 25

d) 4

View Answer

Explanation: Given,

log

_{5}(z

^{2}+ z) – log

_{5}(z + 1) = 2

➩ log

_{5}((z

^{2}+ z) / (z + 1)) = 2

➩ log

_{5}(z(z + 1) / (z + 1)) = 2

➩ log

_{5}z = 2

➩ z = 5

^{2}

➩ 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) p^{2} = q

c) p = q / 2

d) q = p / 3

View Answer

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 = p

^{2}+ q

^{2}+ 2pq

➩ P

^{2}+ q

^{2}+ – 2pq = 0

➩ (p – q)

^{2}= 0

➩ p – q = 0

➩ p = q

10. What will be the value of log_{10} (p^{x}q^{y}), given log_{10}p = x and log_{10}q = y?

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

b) xy^{2}

c) 3x + y^{3}

d) 3xy / 4

View Answer

Explanation: Given,

log

_{10}p = x and log

_{10}q = y

log

_{10}(p

^{x}q

^{y}) = log

_{10}p

^{x}+ log

_{10}q

^{y}

= xlog

_{10}p + ylog

_{10}q

= x

^{2}+ y

^{2}

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