Sure, you could set up any syslog receiver stack like Splunk (as the other OP suggested) or an ELK Stack or even just syslog-ng or rsyslog to disk. Anything that can ingest syslog format will handle Unifi logs.
Decide how you want to receive, store and parse your logstream data. Once you have a syslog receiver set up, set Unifi (System > Site > Enable Remote Logging) for the Syslog server remote address:port and start shipping logs.
Whatever you do with those logs is out of scope for this discussion, but your logger should at least ingest them and spool them.