Answer
The domain is
$$\mathcal{D}=\left\{(x,y,z)\left|\frac{x^2}{2^2}+\frac{y^2}{2^2}+\frac{z^2}{4^2}<1\right.\right\}$$
and it is presented in the figure below
Work Step by Step
The argument of the logarithm has to be positive so we need that $16-4x^2-4y^2-z^2>0$ which can be rewritten as
$$4x^2+4y^2+z^2<16.$$
Dividing this by $16$ we get
$$\frac{x^2}{4}+\frac{y^2}{4}+\frac{z^2}{16}<1$$ and this is the same as
$$\frac{x^2}{2^2}+\frac{y^2}{2^2}+\frac{z^2}{4^2}<1$$
so the domain is, geometrically, the interior of the ellipsoid with the semiaxes of $2,$ $2,$ and $4$ (this is a, so called, prolate spheroid).
So we write for the domain
$$\mathcal{D}=\left\{(x,y,z)\left|\frac{x^2}{2^2}+\frac{y^2}{2^2}+\frac{z^2}{4^2}<1\right.\right\}$$
and it is shown on the figure below
