ClickHouse logs to /var/log/clickhouse-server/clickhouse-server.log on Linux.
Paths are printed exactly as ClickHouse 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.