College Algebra, Chapter 5, Review Exercise, Section Review Exercise, Problem 16

Sketch the graph of the function $f(x) = \ln (x^2)$. State the domain, range and asymptote
By using the properties of Logarithm,

$
\begin{equation}
\begin{aligned}
f(x) &= \ln (x)^2\\
\\
&= 2 \ln (x)
\end{aligned}
\end{equation}
$

The function $g(x) = \ln (x)$ has a vertical asymptote at $x = 0$, domain at $(0, \infty)$ and range of $(-\infty,\infty)$. The graph of $f$ is obtained by
stretching the graph of $g$ 2 units vertically. Since the given function is symmetric to the $y$-axis, then $g$ is also reflected about the $y$-axis. Therefore,
the domain is now $(-\infty,\infty) \bigcup (0, \infty)$. But the range and the vertical asymptote remains unchanged at $(-\infty,\infty)$ and $x = 0$ respectively.