Letting f(x, y) = In($16x^2$ + $y^2$ - 16),
I found the domain to be $16x^2$ + $y^2$ > 16, but I'm having trouble trying to visualise the original function in order to get the range.
Based on tutorials I've seen and my understanding of logarithmic functions, I'm guessing the range, f(x, y) or z, will be all possible values for z?
The range will be the set of real numbers. To see this just take $x=1$. So $f(1,y)=2*ln(|y|)$ and from here it is easy to see why the range is the set of real numbers.