Monty,
You are legally obliged to log your hours as required by the (ICAO member) state that authorizes the operation, and issued the license on which you are flying.
If it is a wet or damp lease, it will (probably) be state under whose rules the operator is registered and has an AOC. Consult your company documentation.
In reality, both the above paragraphs have the same meaning.
To log hours otherwise than in compliance with the rules where you are flying is criminal fraud, and not view highly by the Airstapo.
You can always keep a second log book (as well), my suggestion if often that you should keep a log book that complies with ICAO Annex 1, then any prospective employers, wherever they may be, have a common base of information.
There is nothing particularly sacrosanct about EASA "rules", apart from being stuck with them (with national variations --- beware the national variations) if the carrier is in an EASA country.