Intune Management Extension logs to C:\ProgramData\Microsoft\IntuneManagementExtension\Logs\ on Windows. These paths were verified for msi installs - a different installer, or a portable copy, can put them somewhere else.
Paths are printed exactly as Intune Management Extension 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.