Godot Engine logs to ~/Library/Application Support/Godot/app_userdata/<project name>/logs/godot.log on macOS.
Paths are printed exactly as Godot Engine writes them. Environment variables like %LOCALAPPDATA% and $XDG_STATE_HOME are never expanded, so a path can be pasted straight into a shell. Where a path differs between installer types or shipping flavours, each one is racked as its own card below - why that happens.