Answer
The domain is given by the set $\{(x,y)|x+y< 4\}$.
The range is given by the set (an interval) $(-\infty,\infty)=\mathbb{R}.$
Work Step by Step
1) Domain
Our function is $f(x,y) = \ln(4-x-y).$ Any argument of $\ln$ function must be strictly positive so we need that $4-x-y>0\Rightarrow x+y< 4$. This means that the domain is the set of all ordered pairs $(x,y)$ such that $x+y< 4$ which we will denote by $\{(x,y)|x+y< 4\}$.
2) Range
The value of $\ln$ function can be any real number $(-\infty, \infty)$. Since we have that for $x+y<4$, the expression $4-x-y$ takes all the values from $0$ to $+\infty$ the whole domain of $\ln$ is covers so the whole range has to be covered as well i.e. the range is given by the interval $(-\infty, \infty).$
