Answer
$\{(x,y)\in \mathbb{R}^{2}\ \ |\ \ $ $x\neq 1$ , $y\neq-1$ $\}$
Work Step by Step
$f$ is defined when the argument of the logarithm function is positive,
$xy+x-y-1 \gt 0$
$x(y+1)=(y+1) \gt 0$
$(x-1)(y+1) \gt 0$
So, the domain of f is the whole plane $\mathbb{R}^{2}$, except the two lines $\left\{\begin{array}{l}
x=1\\
y=-1
\end{array}\right.$
(Being excluded, the lines are graphed with dashed lines.)
Domain: $\{(x,y)\in \mathbb{R}^{2}\ \ |\ \ $ $x\neq 1$ , $y\neq-1$ $\}$
