Answer
Domain: $(-\infty,-2)\cup(6,\infty)$
Work Step by Step
Logarithmic functions are defined for positive arguments only.
$ f(x)=\ln(x^{2}-4x-12)$ is defined for positive values.
$ x^{2}-4x-12 \gt 0\quad $
Factor the trinomial -- find factors of $-12$ whose sum is $-4$; we find $-6$ and $+2.$
$(x+2)(x-6) \gt 0$
The graph of $ y=x^{2}-4x-12=(x+2)(x-6)$
is a parabola opening upwards, intersecting the x-axis at $-2$ and $+6.$ We know that $ y $ is positive where the graph is above the x-axis.
The graph is above the x-axis before the left x-intercept $(x \lt -2) $ and to the right of the right intercept $(x \gt 6)$
Thus, the domain is: $(-\infty,-2)\cup(6,\infty)$
