How to tell if a logbook is genuine
Start with what a real history looks like, because once you've seen a good one, a bad one stands out.
A genuine service book has a stamp for each service, and behind most of those stamps sits a tax invoice for the same date, the same odometer reading, and the work that was done. The dates move forward in a sensible pattern. The readings climb at a steady rate that fits how far the car is driven each year. On newer cars, some or all of this may sit in a digital service record held by the maker or dealer instead of a paper book.
The book should also make sense for the car in front of you. A five-year-old car driven 15,000 km a year should show roughly five services, spaced about a year apart. If the story the book tells matches the car, that's a good sign. If it doesn't, that's where you start asking questions.