Let $f$ be a non-vanishing holomorphic function in $z_1,\dots, z_n.$ Suppose $\operatorname{dom}(f)$ is open and connected. Then $\log(f)$ is a holomorphic function.
$\textbf{Q:}$ I can see that $\log(f)$ is holomorphic by considering $\log\circ f(z_1,\dots, z_n)$ and applying chain rule on taking derivatives. Is there a way to define a multivariable version log function by considering $\int_{z_0}^z \frac{df}{f}$ as $f$ is non-vanishing where $z_0,z\in \operatorname{dom}(f)$ and $z_0$ is fixed?
It is not totally clear that given the above definition, the path integral is independent of path, though I can approximate it by rectifiable curves and then I have to shrink the curve. I think I need higher dimensional analogue of Goursat theorem.
If there is a loop $\gamma$ in $\operatorname{dom}(f)$ such that $f\circ \gamma$ has positive winding number with respect to $0$, then you don't really "I can see holomorphicity of $\log(f)$".