Continuous logarithm lifting criterion ($f=e^g⟺\mathrm{wind}(f\circ \gamma ,0)=0\text{ for every loop }\gamma$)

On a locally path-connected space, on each path component choose a value of a complex logarithm of $f$ at a base point. Continue it along paths using local logarithms. Independence of the path is exactly vanishing of the winding number of a continuous closed path for all loops. This constructs a continuous function $g$ with $f=e^g$. The implication from a logarithm to zero winding requires no local connectivity. For applications to arbitrary planar compact sets, apply the criterion on polygonal neighborhoods rather than assuming the compact set is locally path-connected.