Question

$f\left(x\right)=\log x^2$ and $\varphi \left(x\right)=2\log x.$
Are both the function equal?

Answer 1

$f\left(x\right)=\log x^2$

For logarithm to be defined, $x^2>0\Rightarrow x\in R$

$D\left(f\right)=R$

$R\left(f\right)=R$

$\varphi \left(x\right)=2\log x$

For logarithm to be defined,$x>0$

$D\left(\varphi \right)=(0,\infty )$

$R\left(\varphi \right)=R$

Since, $D\left(f\right)\ne D\left(\varphi \right)$

Hence, the two functions are not equal