Practice MCQ Questions and Answer on Logarithm

1.

The value of $$\frac{{6{\text{ }}{{\log }_{10}}1000}}{{3{\text{ }}{{\log }_{10}}100}}$$   is equal to -

• (A) 0
• (B) 1
• (C) 2
• (D) 3

Show Answer

 $$\eqalign{ & \frac{{6{\text{ }}{{\log }_{10}}1000}}{{3{\text{ }}{{\log }_{10}}100}} \cr & = \frac{{6{\text{ }}{{\log }_{10}}{{10}^3}}}{{3{\text{ }}{{\log }_{10}}{{10}^2}}} \cr & = \frac{{6 \times 3{\text{ }}{{\log }_{10}}10}}{{3 \times 2{\text{ }}{{\log }_{10}}10}} \cr & = \frac{{18}}{6} \cr & = 3 \cr} $$

2.

If $${\log _2}\left[ {{{\log }_3}\left( {{{\log }_2}x} \right)} \right] = 1,$$     then x is equal to = ?

• (A) 0
• (B) 12
• (C) 128
• (D) 512

Show Answer

 $$\eqalign{ & {{\log }_2}\left[ {{\text{ }}{{\log }_3}\left( {{\text{ }}{{\log }_2}x} \right)} \right] = 1 \cr & \Rightarrow {\log _3}\left( {{{\log }_2}x} \right) = {2^1} = 2 \cr & \Rightarrow {\log _2}x = {3^2} = 9 \cr & \Rightarrow x = {2^9} = 512 \cr} $$

3.

$$\frac{{\log \sqrt 8 }}{{\log 8}}\,\,{\text{is equal to = ?}}$$

• (A) $$\frac{1}{{\sqrt 8 }}$$
• (B) $$\frac{1}{4}$$
• (C) $$\frac{1}{2}$$
• (D) $$\frac{1}{8}$$

Show Answer

 $$\eqalign{ & \frac{{\log \sqrt 8 }}{{\log 8}} \cr & = \frac{{\log {{\left( 8 \right)}^{\frac{1}{2}}}}}{{\log 8}} \cr & = \frac{{\frac{1}{2}\log 8}}{{\log 8}} \cr & = \frac{1}{2} \cr} $$

4.

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) $${\text{1 + }}\frac{1}{{2!}} + \frac{1}{{3!}} + \frac{1}{{4!}}$$  

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

5.

If $${\log _{10000}}x = - \frac{1}{4}{\text{,}}$$    then the value of x is = ?

• (A) $$\frac{1}{{10}}$$
• (B) $$\frac{1}{{100}}$$
• (C) $$\frac{1}{{1000}}$$
• (D) $$\frac{1}{{10000}}$$

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 $$\eqalign{ & {\text{let }}{\log _{10000}}x = - \frac{1}{4} \cr & \Rightarrow x = {\left( {10000} \right)^{ - \frac{1}{4}}} \cr & = {\left( {{{10}^4}} \right)^{ - \frac{1}{4}}} \cr & = {10^{ - 1}} \cr & = \frac{1}{{10}} \cr} $$

6.

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

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

7.

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

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

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

8.

$${\frac{1}{{\left( {{{\log }_a}bc} \right) + 1}} + }$$   $${\frac{1}{{\left( {{{\log }_b}ca} \right) + 1}} + }$$   $${\frac{1}{{\left( {{{\log }_c}ab} \right) + 1}}}$$   is equal to -

• (A) 1
• (B) $$\frac{3}{2}$$
• (C) 2
• (D) 3

Show Answer

 $$\eqalign{ & {\text{Given}}\,\,\,{\text{Expression}} \cr & = \frac{1}{{{{\log }_a}bc + {{\log }_a}a}} + \frac{1}{{{{\log }_b}ca + {{\log }_b}b}} + \frac{1}{{{{\log }_c}ab + {{\log }_c}c}} \cr & = \frac{1}{{{{\log }_a}\left( {abc} \right)}} + \frac{1}{{{{\log }_b}\left( {abc} \right)}} + \frac{1}{{{{\log }_c}\left( {abc} \right)}} \cr & = {\log _{abc}}a + {\log _{abc}}b + {\log _{abc}}c \cr & = {\log _{abc}}\left( {abc} \right) \cr & = 1 \cr} $$

9.

$$2{\log _{10}}^5 + $$  $${\log _{10}}8 \,- $$  $$\frac{1}{2}{\log _{10}}4$$   = ?

• (A) 2
• (B) 4
• (C) $${\text{2 - 2 lo}}{{\text{g}}_{10}}^2$$
• (D) $${\text{4 - 4 lo}}{{\text{g}}_{10}}^2$$

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 $$\eqalign{ & 2{\log _{10}}5 + {\log _{10}}8 - \frac{1}{2}{\log _{10}}4 \cr & = {\log _{10}}\left( {{5^2}} \right) + {\log _{10}}8 - {\log _{10}}\left( {{4^{\frac{1}{2}}}} \right) \cr & = {\log _{10}}25 + {\log _{10}}8 - {\log _{10}}2 \cr & = {\log _{10}}\left( {\frac{{25 \times 8}}{2}} \right) \cr & = {\log _{10}}100 \cr & = 2 \cr} $$

10.

If $${\log _3}x + {\log _{9}}{x^2} + {\log _{27}}{x^3}$$     $$ = 9,$$  then x equals to -

• (A) 3
• (B) 9
• (C) 27
• (D) None of these

Show Answer

 $$\eqalign{ & \Rightarrow {\log _3}x + {\log _9}{x^2} + {\log _{27}}{x^3} = 9 \cr & \Rightarrow {\log _3}x + {\log _{{3^2}}}{x^2} + {\log _{{3^3}}}{x^3} = 9 \cr & \Rightarrow {\log _3}x + \frac{2}{2}{\log _3}x + \frac{3}{3}{\log _3}x = 9 \cr & \Rightarrow 3{\log _3}x = 9 \cr & \Rightarrow {\log _3}x = 3 \cr & \Rightarrow x = {3^3} = 27 \cr} $$