Logbooks are not required to be carried full stop because no regulation requires their carriage. You just need to produce them if asked - same as your driver's license.
Sometimes we are asked (AOC op) by customers for a copy of the pilot's license - besides the fact they wouldn't know one if they fell over it - I usually answer that they would also require a copy of the medical, log books for recency, FTL records, emergency training records, first aid training, fire fighting training, route check, line check, annual tech exam, a copy of the examiners license who did the OPC, etc etc etc etc - to check if the pilot is legal ................ but we don't carry those around either. We get a couple of official CAA flight audits a year - they cover everything - stickers for limitations in the right place etc etc but never pilot log books.
Of course, if you find it more convenient to carry it for one reason or another then fill your boots but it's not a legal requirement.
best