Answer
$\{(x,y)\in \mathbb{R}^{2}\ \ |\ \ $ $x^{2}+y^{2} \gt 4$ $\}$
Work Step by Step
$f$ is defined when the argument of the logarithm function is positive,
$x^{2}+y^{2}-4 \gt 0$
$x^{2}+y^{2} \gt 4$
This region is bordered by the circle about the origin of radius 2.
The circle itself is excluded (graph with dashed line).
The test point (0,0) does not satisfy the inequality, so the domain is:
the part of the plane outside the circle $x^{2}+y^{2} = 4$
or, in set notation, $\{(x,y)\in \mathbb{R}^{2}\ \ |\ \ $ $x^{2}+y^{2} \gt 4$ $\}$
