Q:

Domain of $f(x,y) = \ln((16-x^2-y^2)(x^2+y^2-4))$

$f(x,y) = \ln((16-x^2-y^2)(x^2+y^2-4))$
I'm stuck in this one because this can be rewritten as:
$$f(x,y) = \ln(16-x^2-y^2) + \ln(x^2+y^2-4)$$
Yet, the domain of the given function is $\{(16-x^2-y^2>0)\land(x^2+y^2-4>0)\} \lor \{(16-x^2-y^2<0)\land (x^2+y^2-4<0)\}$. But the domain of the rewritten on is only $\{(16-x^2-y^2>0)\land(x^2+y^2-4>0)\}$. Which one is the correct one and why does this happen?

A:

Note that $\{(x,y):16-x^2+y^2<0\}\cap\{(x,y):x^2+y^2-4<0\}=\emptyset$, and hence the two domains are the same in different expressions!

