Sharing The Domain?

A simpler description:

We know that we can only take the $\log$ of positive numbers $x>0$ . Therefore:

The domain of $f(x) = 2 \log x$ is $0<x<\infty$ or $(0,\infty)$ , whichever notation you prefer.

Now, remember that $x^2 \geq 0$ for all real values of $x$

This means that besides $x=0$ , any other value of $x$ will give us a positive value for $x^2$ , whose $\log$ can be calculated. Therefore:

The domain of $f(x) = \log(x^2)$ is $-\infty<x<\infty, x \neq 0$ or $(-\infty,0)\cup(0,\infty)$

Therefore, the domains of $f(x) = 2 \log x$ and $f(x) = \log(x^2)$ are not the same, and the answer is $\boxed{\text{No}}$

$\displaystyle log(a \cdot b) \Rightarrow log(|a|) + log(|b|) , \text{ for a, b} \in \mathbb R_{+}$ Note that the condition does not hold both ways and the answer is evident.