$\newcommand{\O}{\mathscr{O}}$Let $X=\Bbb{C}$. Define $\O_X$ to be the sheaf of holomorphic functions, and $\O_X^*$ to be the sheaf of invertible (i.e. nonvanishing) holomorphic functions, the latter of which we consider as an abelian group under multiplication. Define $\exp:\O_X\to\O_X^*$ by $\exp(U)(f)(z)=\exp(f(z))$ for $f\in\O_X(U)$, $z\in\Bbb{C}$.
I'm trying to show that $\exp$ is an epimorphism in the category of sheaves on $X$. This is equivalent to showing that for all $p\in X$, the induced map on stalks $\exp_p:\O_{x,p}\to\O_{x,p}^*$ is a surjection as a set map. This means that given any $g_p\in\O_{x,p}^*(U)$, there exists $f_p\in\O_{x,p}$ such that $g_p=\exp_p(U)(f_p)=\exp(U)(f)_p=\exp(f)_p$.
This amounts to saying that every nonvanishing function has a "local logarithm". I don't know how to show this is the case though, using the fact that $g_p$ is nonvanishing on $U$. Any hints would be greatly appreciated.
(if it matters at all, I'm using Vakil's notes.)