Although I have not flown for a few years, unless something has radically changed logging your hours in your log book has nothing to do with the legality of the aircraft. You would still log even if you had not done your job as a P1 correctly e.g. checking the aircraft documents etc.
I would go with Sam that if you steal an aircraft you can still put the hours in your log book as P1. Might not be the brightest thing in the world to do though!
I would pose another question that could you log a flight as P1 if your licence or your validation was expired by a few days?

People are getting quite wound up on here about logging illegal flights.

You can put whatever you want in your logbook, you could log passenger time in the back of an Easyjet for all it matters, what is relevant is what you are submitting to the authority when applying for a licence.

If you are trying to use hours flown illegally (ie no insurance, stolen aircraft, no ARC, whatever) to obtain a CPL(A), do you think the CAA will issue the licence?

Once pilots have their final licence, they may never send their logbooks off again, if PPL(A) is all you'll achieve, then you could fill your book with illegal fights and unless you happen to get checked by an examiner or audited for some reason, then noone is any the wiser.

Once you submit it for a CPL application, it will be scrutinised, and if there is an audit with no paper trail that you were actually PIC on a legal flight which you are trying to claim towards the CPL requirements, some difficult questions will be asked.

Despite the question as asked by the OP (can I log the time?) That's not the issue, as has-been pointed out in numerous replies. The issue is that the poster cannot legally be P1.

Part FCL Definitions:
Part NCO:
There is no requirement to sign the Journey Log so that is not an issue. So long as you are qualified and "designated" there is no problem.

'wot Whopity said.

But do check the insurance position.

G

Thanks for this ... it is informative and almost certainly more up to date than my last foray into the depths of this subject, it looks like things have moved on ...

Thanks for the detailed and informative post, I (and I am sure others) appreciate the input.

Happy to help. The pace of change over the last few years has not made life especially easy for the industry, relying as it should on standardisation and guidance from the top down. It is interesting to see that this relatively simple/innocuous thread on law versus group practice has input (so far) from at least 3 of us who teach or examine FI courses and try and pass the changes on every day as our role in training the future generations in GA!