Practice MCQ Questions and Answer on Logarithm

1.

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

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

2.

If $$\log x-5\log 3=-2,$$     then x equals -

• (A) 0.8
• (B) 0.81
• (C) 1.25
• (D) 2.43

Show Answer

 $$\eqalign{ & \log x - 5\log 3 = - 2 \cr & \Rightarrow \log x - \log {3^5} = - 2 \cr & \Rightarrow \log \left( {\frac{x}{{{3^5}}}} \right) = - 2 \cr & \Rightarrow \frac{x}{{243}} = {10^{ - 2}} = \frac{1}{{100}} \cr & \Rightarrow x = \frac{{243}}{{100}} = 2.43 \cr} $$

3. 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

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

4.

$$\text{If}\text{log}\frac{a}{b}+\text{log}\frac{b}{a}=\text{log}(a+b),$$      then:

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

Show Answer

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

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) $$\frac{1}{10000}$$

Show Answer

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

6. 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

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

7.

Determine the value of $$\text{lo}{\text{g}}_{3\sqrt{2}}\left(\frac{1}{18}\right)$$   is = ?

• (A) 2
• (B) -2
• (C) $$\sqrt{2}$$
• (D) $$\sqrt{3}$$

Show Answer

 $$\eqalign{ & {\text{lo}}{{\text{g}}_{3\sqrt 2 }}\left( {\frac{1}{{18}}} \right)\, \cr & = {\text{lo}}{{\text{g}}_{3\sqrt 2 }}\left( {\frac{1}{{{{\left( {3\sqrt 2 } \right)}^2}}}} \right)\, \cr & = {\text{lo}}{{\text{g}}_{3\sqrt 2 }}{\text{ }}{\left( {3\sqrt 2 } \right)^{ - 2}} \cr & = \left( { - 2} \right){\text{lo}}{{\text{g}}_{3\sqrt 2 }}3\sqrt 2 \cr & = - 2 \cr} $$

8. 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

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

9.

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

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

10.

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

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