Answer
$\{(x,y)\in \mathbb{R}^{2}\ \ |\ \ $ $x^{2}+y^{2}\lt 4,\ \ x^{2}+y^{2}\neq 3 \}$
Work Step by Step
First, the argument of ln must be positive, $4-x^{2}-y^{2} \gt 0$.
$4 \gt x^{2}+y^{2}$
is the inside of a circle about the origin, radius 2.
(The circle itself is excluded, graphed with a dashed line.)
Second, the denominator must not be zero, and since $\ln 1=0$ we exclude $(x,y)$ for which
$4-x^{2}-y^{2}=1$
$3=x^{2}+y^{2}$
Any points on the circle with radius $\sqrt{3}$ are to be excluded (the circle is graphed with a dashed line).
Domain: $\{(x,y)\in \mathbb{R}^{2}\ \ |\ \ $ $x^{2}+y^{2}\lt 4,\ \ x^{2}+y^{2}\neq 3 \}$
