Answer
The domain is
$$\mathcal{D}=\{(x,y)|x<2,x^2+y^2\neq1\},$$
and it is presented as a blue region on the graph below with red lines excluded from it.
Work Step by Step
We have two constraints:
1. The argument of the logarithm has to be positive i.e.
$$2-x>0\Rightarrow x<2.$$
2. The denominator has to be different than zero i.e.
$$1-x^2-y^2\neq0\Rightarrow x^2+y^2\neq1$$
The first constraint says that we count in only the region of the $xy$ plane that is left from the vertical line $x=2$, excluding the point of this line. The second constraint says that we have to exclude the points from the circle $x^2+y^2=1$ i.e. the circle with the center at the origin and the radius of $1$.
So the domain is
$$\mathcal{D}=\{(x,y)|x<2,x^2+y^2\neq1\},$$
and it is shown on the graph below (the red circle and the red line are excluded from the domain)
