Practice MCQ Questions and Answer on Logarithm

31.

What is the value of the following expression?
$$\log \left( {\frac{9}{{14}}} \right) - $$   $$\log \left( {\frac{{15}}{{16}}} \right) + $$   $$\log \left( {\frac{{35}}{{24}}} \right)$$

• (A) 0
• (B) 1
• (C) 2
• (D) 3

Show Answer

 $$\eqalign{ & \log \left( {\frac{9}{{14}}} \right) - \log \left( {\frac{{15}}{{16}}} \right) + \log \left( {\frac{{35}}{{24}}} \right) \cr & = \log \left( {\frac{9}{{14}} \div \frac{{15}}{{16}} \times \frac{{35}}{{24}}} \right) \cr & = \log \left( {\frac{9}{{14}} \times \frac{{16}}{{15}} \times \frac{{35}}{{24}}} \right) \cr & = \log 1 \cr & = 0 \cr} $$

32.

The value of log₂ 16 is:

• (A) $$\frac{1}{8}$$
• (B) 4
• (C) 8
• (D) 16

Show Answer

 Let log2 16 = n. Then, 2n = 16 = 24 ⇒ n = 4 ∴ log2 16 = 4

33.

If $$\log \frac{a}{b} + \log \frac{b}{a} = $$   $$\,\log \left( {a + b} \right),$$   then -

• (A) a + b = 1
• (B) a - b = 1
• (C) a = b
• (D) a2 - b2 = 1

Show Answer

 $$\eqalign{ & \log \frac{a}{b} + \log \frac{b}{a} = \log \left( {a + b} \right) \cr & \Rightarrow \log \left( {a + b} \right) = \log \left( {\frac{a}{b} \times \frac{b}{a}} \right) \cr & \Rightarrow \log \left( {a + b} \right) = \log 1 \cr & So,\,\,\,a + b = 1 \cr} $$

34.

If $${\log _{10}}5 + {\log _{10}}\left( {5x + 1} \right)$$     = $${\log _{10}}$$ $$\left( {x + 5} \right)$$ $$\, + $$ $$1,$$ then x is equal to :

• (A) 1
• (B) 3
• (C) 5
• (D) 10

Show Answer

 $$\eqalign{ & \Rightarrow {\log _{10}}5 + {\log _{10}}\left( {5x + 1} \right) = {\log _{10}}\left( {x + 5} \right) + 1 \cr & \Rightarrow {\log _{10}}\left[ {5\left( {5x + 1} \right)} \right] = {\log _{10}}\left[ {10\left( {x + 5} \right)} \right] \cr & \Rightarrow 5\left( {5x + 1} \right) = 10\left( {x + 5} \right) \cr & \Rightarrow 5x + 1 = 2x + 10 \cr & \Rightarrow 3x = 9 \cr & \Rightarrow x = 3 \cr} $$

35.

If $${\log _{10}}125 + {\log _{10}}8 = x,$$     then x is equal to -

• (A) $$\frac{1}{3}$$
• (B) 0.064
• (C) -3
• (D) 3

Show Answer

 $$\eqalign{ & {\log _{10}}125 + {\log _{10}}8 = x \cr & \Rightarrow {\log _{10}}\left( {125 \times 8} \right) = x \cr & \Rightarrow x = {\log _{10}}\left( {1000} \right) \cr & \Rightarrow x = {\log _{10}}{\left( {10} \right)^3} \cr & \Rightarrow x = 3{\log _{10}}10 \cr & \Rightarrow x = 3 \cr} $$

36.

If log_x y = 100 and log₂ x = 10, then the value of y is:

• (A) 210
• (B) 2100
• (C) 21000
• (D) 210000

Show Answer

 $$\eqalign{ & {\log _2}x = 10\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,x = {2^{10}} \cr & \therefore {\log _x}y = 100 \cr & \Rightarrow y = {x^{100}} \cr & \Rightarrow y = {\left( {{2^{10}}} \right)^{100}}\,\,\,\left[ {{\text{put}}\,{\text{value}}\,{\text{of}}\,x} \right] \cr & \Rightarrow y = {2^{1000}} \cr} $$

37. If $$\log 2 = 0.30103,$$    the number of digits in $${4^{50}}$$ is -

• (A) 30
• (B) 31
• (C) 100
• (D) 200

Show Answer

 $$\eqalign{ & \log {4^{50}} \cr & = 50\log 4 \cr & = 50\log {2^2} \cr & = \left( {50 \times 2} \right)\log 2 \cr & = 100 \times \log 2 \cr & = \left( {100 \times 0.30103} \right) \cr & = 30.103 \cr & \therefore {\text{characteristic}} = 30, \cr} $$ Hence, the number of digits in $${{\text{4}}^{50}} = 31$$

38. If $${\log _{10}}a = p,$$   $${\log _{10}}b = q,$$   then what is $${\log _{10}}\left( {{a^p}{b^q}} \right)$$   equal to?

• (A) p2 + q2
• (B) p2 - q2
• (C) p2q2
• (D) $$\frac{{{p^2}}}{{{q^2}}}$$

Show Answer

 $$\eqalign{ & {\text{Given}}, \cr & {\log _{10}}a = p,\,{\log _{10}}b = q \cr & {\log _{10}}\left( {{a^p}{b^q}} \right) = {\log _{10}}{a^p} + {\log _{10}}{b^q} \cr & = p{\log _{10}}a + q{\log _{10}}b \cr & = {p^2} + {q^2} \cr} $$

39.

If $$a = {b^2} = {c^3} = {d^4},$$    then the value of $${\log _a}\left( {abcd} \right)$$   would be -

• (A) $${\log _a}1 + {\log _a}2 + {\log _a}3 + {\log _a}4$$      
• (B) $${\log _a}24$$
• (C) $${\text{1 + }}\frac{1}{2} + \frac{1}{3} + \frac{1}{4}$$  
• (D) $${\text{1 + }}\frac{1}{{2!}} + \frac{1}{{3!}} + \frac{1}{{4!}}$$  

Show Answer

 $$\eqalign{ & a = {b^2} = {c^3} = {d^4} \cr & \Rightarrow b = {a^{\frac{1}{2}}},\,\,\,\,c = {a^{\frac{1}{3}}},\,\,\,\,d = {a^{\frac{1}{4}}} \cr & \therefore {\log _a}\left( {abcd} \right) \cr & = {\log _a}\left( {a \times {a^{\frac{1}{2}}} \times {a^{\frac{1}{3}}} \times {a^{\frac{1}{4}}}} \right) \cr & = {\log _a}{a^{\left( {1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4}} \right)}} \cr & = \left( {1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4}} \right){\log _a}a \cr & = 1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} \cr} $$

40.

If log 2 = 0.30103, the number of digits in 2⁶⁴ is:

• (A) 18
• (B) 19
• (C) 20
• (D) 21

Show Answer

 $$\eqalign{ & \log \left( {{2^{64}}} \right) \cr & = 64 \times \log 2 \cr & = \left( {64 \times 0.30103} \right) \cr & = 19.26592 \cr} $$ Its characteristic is 19. Hence, then number of digits in 264 is 20.