Answer
The domain is given by the set $\{(x,y)|xy>6\}$.
The range is given by the set (an interval) $(-\infty,\infty).$
Work Step by Step
1) Domain
The given function is $f(x,y)=\ln(xy-6)$. Any argument of $\ln$ must not be negative so we need that $xy-6>0\Rightarrow xy>6,$ so the domain is the set of all ordered pairs $(x,y)$ such that $xy>6$ which we will denote by $\{(x,y)|xy>6\}$.
2) Range
The value of $\ln$ function can be any real number $(-\infty,\infty)$. We can always take $x$ and $y$ from the domain so that the argument $xy-6$ takes any value from $0$ to $\infty$ i.e. the whole domain of $\ln$ is covered which means that the whole range has to be covered as well. So the range of $f(x,y)$ is $(-\infty,\infty).$
