$$\text{The argument of the log must be positive:}$$ $$\ 16-4x^2-4y^2-z^2>0$$ $$\ 4x^2+4y^2+z^2<16$$ $$\ \frac{x^2}{4}+\frac{y^2}{4}+\frac{z^2}{16}<1$$ $$\ (\frac{x}{2})^2+(\frac{y}{2})^2+(\frac{z}{4})^2<1$$ $$\text {so the domain is a region enclosed in an ellipsoid.}$$ $$\ \\\ \text{Now you calculate what happens to the range 'moving in the domain':}$$ $$\text{note that }$$ $$\ log(16-4x^2-4y^2-z^2)=log[16-(4x^2+4y^2+z^2)]=log[16-p(x,y,z)]$$ $$\ \text{and }\ p(x,y,z)≥0 \forall(x,y,z).$$ $$\text{In the origin }p(x,y,z)=0 \text{ so you have the maximum value in the range: }$$ $$\ f(x,y,z)_{max}=log16$$ $$\text{Approaching the surface of the ellipsoid instead, the argument of the log tends to }\ 0^+$$ $$\text{so }\ f(x,y,z)\to -\infty$$ $$\text{The total range should be therefore: }\ (-\infty,log16]$$