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Let me introduce you to syslogd.
But well, it’s probably overkill, and you almost certainly just need to log on a shared volume.
Syslog isn’t really overkill IMO. It’s pretty easy to configure it to log to a remote server, and to split particular log types or sources into different files. It’s a decent abstraction - your app that logs to syslog doesn’t have to know where the logs are going.