Practice MCQ Questions and Answer on Logarithm

1. If the logarithm of a number is - 3.153, what are characteristic and mantissa?

• (A) characteristic = -4, mantissa = 0.847
• (B) characteristic = -3, mantissa = -0.153
• (C) characteristic = 4, mantissa = -0.847
• (D) characteristic = 3, mantissa = -0.153

Show Answer

 $$\eqalign{ & {\text{let }}\log x = - 3.153 \cr & {\text{then, }}\log x = - 3.153 \cr & = - 3 + \left( { - 0.153} \right) \cr & = \left( { - 3 - 1} \right) + \left( {1 - 0.153} \right) \cr & = - 4 + 0.847 = \overline 4 .847 \cr & {\text{Hence,}} \cr & {\text{characteristic = - 4, mantissa = 0}}{\text{.847 }} \cr} $$

2.

$$\frac{\log \sqrt{8}}{\log 8}$$  is equal to:

• (A) $$\frac{1}{6}$$
• (B) $$\frac{1}{4}$$
• (C) $$\frac{1}{2}$$
• (D) $$\frac{1}{8}$$

Show Answer

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

3. If $$\log 2=0.3010$$   and $$\log 3=0.4771,$$   the value of $${\log }_{5}512$$   = ?

• (A) 2.870
• (B) 2.967
• (C) 3.876
• (D) 3.912

Show Answer

 $$\eqalign{ & {\log _5}512 \cr & = \frac{{\log 512}}{{\log 5}} \cr & = \frac{{\log {2^9}}}{{\log \left( {\frac{{10}}{2}} \right)}} \cr & = \frac{{9\log 2}}{{\log 10 - \log 2}} \cr & = \frac{{\left( {9 \times 0.3010} \right)}}{{1 - 0.3010}} \cr & = \frac{{2.709}}{{0.699}} \cr & = \frac{{2709}}{{699}} \cr & = 3.876 \cr} $$

4. If $$\text{a}=\text{lo}{\text{g}}_{\text{8}}\text{225}$$   and $$\text{b = lo}{\text{g}}_{\text{2}}\text{15},$$   then a in terms of b is -

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

Show Answer

 $$\eqalign{ & a = {\log _8}225 \cr & = {\log _{{2^3}}}\left( {{{15}^2}} \right) \cr & = \frac{2}{3}{\log _2}15 \cr & = \frac{{2b}}{3} \cr} $$

5.

$$\frac{1}{{\log }_{a}b}\times \frac{1}{{\log }_{b}c}\times \frac{1}{{\log }_{c}a}$$     is equal to -

• (A) a + b + c
• (B) abc
• (C) 0
• (D) 1

Show Answer

 $$\eqalign{ & {\text{Given}}\,\,\,{\text{Expression}} \cr & \left( {\frac{{\log a}}{{\log b}} \times \frac{{\log b}}{{\log c}} \times \frac{{\log c}}{{\log a}}} \right) \cr & = 1 \cr} $$

6.

If log 2 = 0.3010 and log 3 = 0.4771, the value of log₅ 512 is:

• (A) 2.870
• (B) 2.967
• (C) 3.876
• (D) 3.912

Show Answer

 $$\eqalign{ & {\log _5}512 \cr & = {{\log 512} \over {\log 5}} \cr & = {{\log {2^9}} \over {\log \left( {{{10} \over 2}} \right)}} \cr & = {{9\log 2} \over {\log 10 - \log 2}} \cr & = {{ {9 \times 0.3010} } \over {1 - 0.3010}} \cr & = {{2.709} \over {0.699}} \cr & = {{2709} \over {699}} \cr & = 3.876 \cr} $$

7. If $$\log 3\log \left(3^x-2\right)$$   and $$\log \left(3^x+4\right)$$   are in arithmetic progression, then x is equal to

• (A) $$\frac{8}{3}$$
• (B) $$\log 3^8$$
• (C) $$\log 2^3$$
• (D) 8

Show Answer

 In arithmetic progression common ratio are equal to $$\log \left( {{3^x} - 2} \right) - \log 3 = $$     $$\log \left( {{3^x} + 4} \right) - $$   $$\log \left( {{3^x} - 2} \right)$$ $$\frac{{\log \left( {{3^x} - 2} \right)}}{{\log 3}} = \frac{{\log \left( {{3^x} + 4} \right)}}{{\log \left( {{3^x} - 2} \right)}}$$     $$\left( {\therefore \log a - \log b = \log \frac{a}{b}} \right)$$ $$\eqalign{ & \frac{{\log {3^x}}}{{\log 2\log 3}} = \frac{{x\log 3\log 4\log 2}}{{x\log 3}} \cr & \frac{{x\log 3}}{{\log 2\log 3}} = \frac{{x\log 3\log 4\log 2}}{{x\log 3}} \cr & \frac{x}{{\log 2}} = \log 4\log 2 \cr & x = \log 4\log \,2\log 2 \cr & x = \log 8 \cr & x = \log {2^3} \cr} $$

8.

If $$\text{lo}{\text{g}}_8\text{p}=25$$   and $$\text{lo}{\text{g}}_2\text{q}=5,$$   then -

• (A) p = q15
• (B) p2 = q3
• (C) p = q5
• (D) p3 = q

Show Answer

 $$\eqalign{ & {\text{lo}}{{\text{g}}_8}{\text{p}} = 25\,\,{\text{and lo}}{{\text{g}}_2}{\text{q}} = 5\, \cr & \Rightarrow {\text{p = }}{{\text{8}}^{25}}\,{\text{and q}} = {2^5} \cr & \Rightarrow {\text{p = }}{\left( {{2^3}} \right)^{25}}{\text{ and q = }}{2^5} \cr & \Rightarrow {\text{p = }}{{\text{2}}^{75}}{\text{ and q = }}{2^5} \cr & \Rightarrow {\text{p = }}{\left( {{2^5}} \right)^{15}}{\text{ and q = }}{2^5} \cr & \Rightarrow {\text{ p = }}{{\text{q}}^{15}} \cr} $$

9.

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

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

10.

If $${\log }_{10}7=a,$$   then $${\log }_{10}\left(\frac{1}{70}\right)$$   is equal to -

• (A) -(1 + a)
• (B) (1 + a)-1
• (C) $$\frac{a}{10}$$
• (D) $$\frac{1}{10a}$$

Show Answer

 $$\eqalign{ & {\log _{10}}\left( {\frac{1}{{70}}} \right) \cr & = {\log _{10}}1 - {\log _{10}}70 \cr & = - {\log _{10}}\left( {7 \times 10} \right) \cr & = - \left( {{{\log }_{10}}7 + {{\log }_{10}}10} \right) \cr & = - \left( {a + 1} \right) \cr} $$