There doesn't seem to be any reasonable generic log facility where these things should live.
I'm not talking about just capturing server errors - which I expect would end up in the server error log, I'm also talking about informational updates (eg "X number of emails sent by this cron entry") and debugging - indeed something which can deal with all of the PSR-3 log levels (emergency, alert, critical, error, warning, notice, info, debug).
Are you logging to a file? Custom DB table?
As an aside - I think a generic activity log (with admin UI) which implements the PSR-3 logging interface would be very useful as part of the core.
Add-ons can then log activity and errors there for the administrators to monitor.
The UI would allow filtering log messages by date, PSR-3 log level and/or add-on id.
We could also potentially extend the capabilities of this core logging facility to send log information to external application monitoring systems.
I might add this as a suggestion.