Yes, they do check your log book, but not as thoroughly as they could.
They check:
- You've added everything up correctly.
- You've logged sufficient hours to fulfill the requirements.
They don't check:
- That your logbook agrees with the aircraft tech log.
- That the aircraft was serviceable for those flights (not in maintenance etc.)
Basically they check, but they don't get Poirot on the case unless they smell a rat.