Holomorphic logarithm when $f'/f$ has zero periods

Let \(\Omega\subseteq\CC\) be a connected open set and let \(f\) be holomorphic and nonvanishing on \(\Omega\). Suppose that \(\int_\gamma\frac{f'(z)}{f(z)}\dz=0\) for every piecewise smooth closed curve \(\gamma\) in \(\Omega\). Fix \(z_0\in\Omega\) and \(c\in\CC\) with \(e^c=f(z_0)\), and define \[ g(z)\coloneqq c+\int_{z_0}^z\frac{f'(\xi)}{f(\xi)}\dxi, \] the integral taken along any piecewise smooth path in \(\Omega\) from \(z_0\) to \(z\). Then \(g\) is well defined and holomorphic on \(\Omega\), \(g'=f'/f\), and \(e^{g}=f\) on \(\Omega\).