Memcached logs to /var/log/memcached.log on Linux. 1 other Linux log location is recorded below.
Paths are printed exactly as Memcached 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.