Answer
Domain: $(-\infty,-5)\cup(2,\infty)$
Work Step by Step
Logarithmic functions are defined for positive arguments only.
$ f(x)=\displaystyle \ln(\frac{x-2}{x+5})$ is defined for $\displaystyle \frac{x-2}{x+5}\gt 0.$
The boundary points are $-5$ and $2.$
Testing each interval $:$
$\left[\begin{array}{llllllll}
& -\infty & & -5 & & 2 & & \infty\\
\text{test point} & & -10 & & 0 & & 10 & \\
f(x) & & \frac{-12}{-15} & & \frac{-2}{5} & & \frac{8}{15} & \\
\text{sign} & & + & & - & & + &
\end{array}\right]$
Domain: $(-\infty,-5)\cup(2,\infty)$ .
