Question

The equivalent function of $ \log {x^2} $ is
A. $ 2\log x $
B. $ 2\log \left| x \right| $
C. $ \left| {\log {x^2}} \right| $
D. $ {(\log x)^2} $

Answer 1

Hint: Here before solving this question we need to know the property of logarithm: -
 $ \log {m^n} = n\log m\,\,\,\,\,\,\,\,\,\,...(1) $
There is no alternate method but to apply the property of log to get the required result.

Complete step-by-step answer:
In mathematics, the logarithm is the inverse function to exponentiation. That means the logarithm of a given number x is the exponent to which another fixed number, the base b, must be raised, to produce that number x.
According to this question we have,
 $ \log {x^2} $
Since we know that logarithm takes only a positive number and we also know that property of $ {x^2} $ is whether there is a positive number or negative number the result is always positive in that case there will be a problem if negative values come into play.
So, in order to avoid the negative sign, we will use the modulus function.
 $ m = x\,{\text{and}}\,n = 2 $
Substitute all the values in equation (1).
 $ \log {x^2} = 2\log \left| x \right| $
So, the correct answer is “Option b”.

Note: In this question, there is a confusion between option a and option b. So in order to eliminate the option, we will use the basic definition of a domain.