Practice MCQ Questions and Answer on Logarithm

31. 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 $${{\text{4}}^{50}} = 31$$

32.

If log₁₀ 2 = 0.3010, then log₂ 10 is equal to:

• (A) $$\frac{699}{301}$$
• (B) $$\frac{1000}{301}$$
• (C) 0.3010
• (D) 0.6990

Show Answer

 $$\eqalign{ & {\log _2}10 \cr & = \frac{1}{{{{\log }_{10}}2}} \cr & = \frac{1}{{0.3010}} \cr & = \frac{{10000}}{{3010}} \cr & = \frac{{1000}}{{301}} \cr} $$

33. The number of digits in $${\text{4}}^9\times {\text{5}}^{17}\text{,}$$   when expressed in usual form, is -

• (A) 16
• (B) 17
• (C) 18
• (D) 19

Show Answer

 $$\eqalign{ & \log \left( {{4^9} \times {5^{17}}} \right) \cr & = \log \left( {{4^9}} \right) + \log \left( {{5^{17}}} \right) \cr & = \log \left( {{2^2}} \right)^9 + \log \left( {{5^{17}}} \right) \cr & = \log \left( {{2^{18}}} \right) + \log \left( {{5^{17}}} \right) \cr & = 18\log 2 + 17\log 5 \cr & = 18\log 2 + 17\left( {\log 10 - \log 2} \right) \cr & = 18\log 2 + 17\log 10 - 17\log 2 \cr & = \log 2 + 17\log 10 \cr & = 0.3010 + 17 \times 1 = 17.3010 \cr & \therefore {\text{Characteristic}} = 17 \cr & {\text{Hence, the number of digits in }}\left( {{4^9} \times {5^{17}}} \right) = 18 \cr} $$

34.

Which of the following statements is not correct?

• (A) log10 10 = 1
• (B) log (2 + 3) = log (2 x 3)
• (C) log10 1 = 0
• (D) log (1 + 2 + 3) = log 1 + log 2 + log 3

Show Answer

 $$\eqalign{ & {\text{A = Since lo}}{{\text{g}}_a}a = 1,{\text{ so lo}}{{\text{g}}_{10}}10 = 1 \cr & \cr & B = \,{\text{log}}\left( {2 + 3} \right) = 5\, \cr & \,\,\,{\text{and log}}\left( {2 \times 3} \right) = {\text{log}}6 \cr & = {\text{log}}2 + {\text{log}}3 \cr & \therefore \,{\text{log}}\left( {2 + 3} \right) \ne {\text{log}}\left( {2 \times 3} \right). \cr & \cr & C = \,{\text{ Since lo}}{{\text{g}}_a}1 = 0,so{\text{ lo}}{{\text{g}}_{10}}1 = 0. \cr & \cr & {\text{D = log}}\left( {1 + 2 + 3} \right) = {\text{ log}}6 \cr & = {\text{ log}}\left( {1 \times 2 \times 3} \right) \cr & = {\text{ log}}1 + {\text{ log}}2 + {\text{ log}}3. \cr & {\text{So (B) is incorrect}} \cr} $$

35.

If a^x = b^y, then:

• (A) $$\log \frac{a}{b}=\frac{x}{y}$$
• (B) $$\frac{\log a}{\log b}=\frac{x}{y}$$
• (C) $$\frac{\log a}{\log b}=\frac{y}{x}$$
• (D) None of these

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 $$\eqalign{ & {a^x} = {b^y} \cr & \Rightarrow \log {a^x} = \log {b^y} \cr & \Rightarrow x\log a = y\log b \cr & \Rightarrow {{\log a} \over {\log b}} = {y \over x} \cr} $$

36.

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

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

37.

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

• (A) 0
• (B) 1
• (C) 2
• (D) 3

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

38.

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

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

39.

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

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

Show Answer

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

40.

The value of $$\text{lo}{\text{g}}_{10}\left(0.0001\right)$$   is = ?

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

Show Answer

 $$\eqalign{ & {\text{lo}}{{\text{g}}_{10}}\left( {0.0001} \right)\, \cr & = {\text{lo}}{{\text{g}}_{10}}\left( {\frac{1}{{10000}}} \right) \cr & \, = {\text{lo}}{{\text{g}}_{10}}\left( {\frac{1}{{{{10}^4}}}} \right) \cr & \, = {\text{lo}}{{\text{g}}_{10}}{10^{ - 4}} \cr & \, = - 4\,{\text{lo}}{{\text{g}}_{10}}10 \cr & = - 4{\text{ }} \cr} $$