Answer
Domain = $\displaystyle \{(x,y,z)|\quad\frac{x^{2}}{4}+\frac{y^{2}}{4}+\frac{z^{2}}{16}\lt 1\}$
(see image)
Work Step by Step
Because of the restriction for logarithmic functions, f is defined only for those points $(x,y,z)\in \mathbb{R}^{3}$ for which
$16-4x^{2}-4y^{2}-z^{2} \gt 0$
$4x^{2}+4y^{2}+z^{2}\lt 16\qquad/\div 16$
$\displaystyle \frac{x^{2}}{4}+\frac{y^{2}}{4}+\frac{z^{2}}{16}\lt 1$
$(0,0,0)$ satisfies the condition, so the domain is the set of points inside
$\displaystyle \frac{x^{2}}{4}+\frac{y^{2}}{4}+\frac{z^{2}}{16}=1\qquad$ (elipsoid)
Domain = $\displaystyle \{(x,y,z)|\quad\frac{x^{2}}{4}+\frac{y^{2}}{4}+\frac{z^{2}}{16}\lt 1\}$
(see image)
