Question

The logarithm of 0.125 to the base 2 is equal to
$
  {\text{A}}{\text{. 4}} \\
  {\text{B}}{\text{. 7}} \\
  {\text{C}}{\text{. - 1}} \\
  {\text{D}}{\text{. - 3}} \\
$

Answer 1

Hint: -To answer this type of question we should have knowledge of properties of logarithm. Here first we will convert 0.125 in to power of 2 and then applying the logarithmic property $\left( {{{\log }_a}{a^m} = m{{\log }_a}a = m} \right)$ we will proceed further.

Complete step-by-step answer:
We have to find
${\log _2}0.125$
As we understand from the hint first we have to convert 0.125 into the power of 2 so that by using properties of logarithm we can proceed further.
Hence we can write 0.125 as
$0.125 = \dfrac{{125}}{{1000}} = \dfrac{1}{8} = \dfrac{1}{{{2^3}}} = {2^{ - 3}}$
Now our required question becomes
${\log _2}{2^{ - 3}}$ and now we will apply the property $\left( {{{\log }_a}{a^m} = m{{\log }_a}a = m} \right)$ on using this property solution of question becomes very easy.
We have ${\log _2}{2^{ - 3}}$
We can rewrite it as ${\log _2}{2^{ - 3}} = - 3{\log _2}2 = - 3$
Hence option D is the correct option.

Note: -Whenever we get this type of question the key concept of solving is we have to understand from the option that if options are in integer means we have to use some properties that can convert this decimal number of logarithm into integer. And hence we should have remembered all the properties of logarithm to solve this type of question easily.