I will be grateful if someone could explain the following statement from p. 26 of the book "Analytic functions of several complex variables" by Gunning and Rossi:
As a function of $\zeta$ for $z$ fixed: $$ d \log|\zeta - z|^2 = d(\log(\zeta-z) + \log(\overline{\zeta}-\overline{z})) = \frac{d\zeta}{\zeta-z}+ \frac{d \overline{\zeta}}{\overline{\zeta}-\overline{z}} $$
On the previous page the exterior differential operator $d = \partial + \overline{\partial}$ has been defined for smooth $(p,q)$-forms. But $\log(w)$ is not even a continuous function for $w\in \mathbb{C}$, how can we apply the exterior derivative operator?
In particular, how does the following operation makes any sense: $$\frac{\partial \log(\zeta-z)}{\partial \zeta} = \frac{1}{\zeta-z}$$
Indeed, logarithm is generally not globally defined. However given an open set $U$ you can always find an open cover $U_{\alpha}$ such that the logarithm can be defined on each $U_{\alpha}$. On each $U_{\alpha}$ you have the relation $d \log z = \frac{dz}{z}$. Clearly the right hand side agrees on the overlaps of $U_{\alpha}$, so it is a globally defined $1$-form (even though the function $\log z$ is not).
In summary, you are right to think that equation $d \log z = \frac{dz}{z}$ is not completely correct. In fact it is a mental shortcut describing what I wrote above.