1. The problem is to find the domain of the function $f(x,y) = \ln(xy)$.\n\n2. The natural logarithm function $\ln(z)$ is defined only for $z > 0$. This means the argument inside the logarithm must be positive.\n\n3. For $f(x,y) = \ln(xy)$, the argument is $xy$. So we require:\n$$xy > 0$$\n\n4. The product $xy$ is positive if both $x$ and $y$ are positive, or both are negative.\n\n5. Therefore, the domain is:\n$$\{(x,y) \in \mathbb{R}^2 : x > 0 \text{ and } y > 0\} \cup \{(x,y) \in \mathbb{R}^2 : x < 0 \text{ and } y < 0\}$$\n\n6. In simpler terms, the domain consists of the first and third quadrants of the $xy$-plane, excluding the axes where $x=0$ or $y=0$ because $xy=0$ is not allowed.