If you must check a 3CX log I'd say check the 3CX System service log for entries starting with "Inserted.REGISTRAR"
Just bear in mind that it might not be that straight forward and I would not generally recommend monitoring this log for registration status as it was not designed for this purpose and you might end up with misinformation.
Note: You'd probably also need to enable Verbose Logging mode which will consume inevitably additional hardware resources.