By the condition of the logarithm, we have: \begin{align*} 16-4x^2-4y^2-z^2 &> 0 \\ 4x^2+4y^2+z^2 &<16 \\ \frac{x^2}{4}+\frac{y^2}{4}+\frac{z^2}{16} &< 1 \end{align*} Therefore the domain of $ g $ is the interior of an origin-centered ellipsoid.
I think this is the correct solution, I await your comments. If anyone has a different solution or correction of my work I will be grateful.
