1. The problem is to find the function $f(x) = \ln x^2$ and understand its properties. 2. Recall that $\ln x^2 = \ln \left(x^2\right)$. 3. We use the logarithm power rule, which states that $\ln a^b = b \ln a$. 4. Applying this rule, we get $f(x) = 2 \ln |x|$, since the logarithm argument must be positive, we write absolute value $|x|$. 5. The domain of $f(x)$ is $x \in (-\infty,0) \cup (0,\infty)$ because $\ln$ is defined for positive numbers and we use $|x|$. 6. The function can be graphed as $y = 2 \ln |x|$ which has a vertical asymptote at $x=0$ and is symmetric about the y-axis. 7. Final function: $$f(x) = 2 \ln |x|$$