Question

The logarithm of 0.0625 to the base 2 is
(a) 0.025
(b) 0.25
(c) 5
(d) $-4$
(e) $-2$

Answer 1

Hint: First, here we will convert decimal form of 0.0625 in fraction form i.e. $\dfrac{625}{10000}$ . We will further reduce this form. On getting the reduced form we will use the property of logarithm as $\log \left( \dfrac{a}{b} \right)=\log a-\log b$ , $\log 1=0$ , ${{\log }_{n}}\left( {{n}^{m}} \right)=m{{\log }_{n}}n$ and ${{\log }_{n}}n=1$ . Thus, we will get the answer.

Complete step-by-step answer:
In the question, we are asked to find a value of logarithm of 0.0625 to the base 2 which is written in mathematical form as ${{\log }_{2}}0.0625$ .
So, first we will convert 0.0625 from decimal form to the fraction form i.e. $\dfrac{625}{10000}$ .
So, taking 25 common from numerator and denominator, we will get $\dfrac{25}{400}$ .
Now, again dividing by 25 in numerator and denominator, we get value as $\dfrac{1}{16}$ .
So, now our question becomes ${{\log }_{2}}\dfrac{1}{16}$ .
Now, we know that $16=2\times 2\times 2\times 2={{2}^{4}}$ . So, substituting this value in place of 16, we get ${{\log }_{2}}\dfrac{1}{{{2}^{4}}}$ .
Now, we will use the division rule i.e. given as $\log \left( \dfrac{a}{b} \right)=\log a-\log b$
$\therefore {{\log }_{2}}\left( \dfrac{1}{{{2}^{4}}} \right)={{\log }_{2}}1-{{\log }_{2}}\left( {{2}^{4}} \right)$
Now, we know that value of $\log 1=0$ and ${{\log }_{n}}\left( {{n}^{m}} \right)=m{{\log }_{n}}n$ . So, using this, we get
${{\log }_{2}}\left( \dfrac{1}{{{2}^{4}}} \right)=0-4{{\log }_{2}}2$
Also, we know that ${{\log }_{n}}n=1$ so, on applying this property we will get
${{\log }_{2}}\left( \dfrac{1}{{{2}^{4}}} \right)=0-4=-4$
Thus, the logarithm of 0.0625 to the base 2 is $-4$ .
Hence, option (d) is the correct answer.

Note: Students might make mistakes while taking base 10 instead of base 2. Due to this, there will be change in answer i.e. given as ${{\log }_{10}}\left( \dfrac{1}{{{2}^{4}}} \right)$ and on solving will get the answer as $-1.2041$ which is incorrect. So, this is a common mistake students make in solving the problem and end up getting the wrong answer. So, please be careful while taking the base of what is given in question and then solve it accordingly.