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

The equivalent function of log⁡x2\log {x^2} is
A. 2log⁡x2\log x
B. 2log⁡∣x∣2\log \left| x \right|
C. ∣log⁡x2∣\left| {\log {x^2}} \right|
D. (log⁡x)2{(\log x)^2}

Explanation

Solution

Hint : Here before solving this question we need to know the property of logarithm: -
log⁡mn=nlog⁡m          ...(1)\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⁡x2\log {x^2}
Since we know that logarithm takes only a positive number and we also know that property of x2{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 and n=2m = x\,{\text{and}}\,n = 2
Substitute all the values in equation (1).
log⁡x2=2log⁡∣x∣\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.