The domain of $f(x)= \log(x) + \log(y)$ Vs the domain of $g(x) = \log(xy)$

Question

Let:

$f(x) = \log(x) + \log(y)$ and $g(x) = \log(xy)$

As we know: $\log(xy) = \log(x) + \log(y)$, so I figure that $f(x) = f(g)$

The domain of $f(x)$ is : $x>0$ and $y>0$.

And the domain of $g(x)$ is: $x>0$ and $y>0$ or $x<0$ and $y<0$.

Why are the two domains different ?

Answer 1

Your statement "$\log(xy)=\log x+\log y$" is incomplete. It should be

if $x,y>0$, then $\log(xy)=\log x+\log y$.

Therefore your functions $f$ and $g$ have equal values whenever $x,y>0$. However this says nothing at all about the functions - whether they are defined, undefined, equal, unequal - for other values of $x$ and $y$. So there is no reason at all why one of the functions cannot have a larger domain than the other.