AdGuard Home logs to /var/log/AdGuardHome.err on Linux. 2 other Linux log locations are recorded below.
Paths are printed exactly as AdGuard Home 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.