You could always take the Tail #'s and cross references them to
www.flightaware.com
Not everything shows up on flightaware, by a long shot. I've looked up aircraft I just flew across the country, and flightaware says the airplane has been sitting somewhere for two months. Further, it won't show VFR flights.
It also won't show blocked flights.
As far as falsifying one's log, there's little doubt that a lot of pilots out there have done exactly that. I worked with an assistant chief pilot years ago who is now flying for an airline somewhere, having jumped rapidly from single Cessnas to corporate aircraft and then to an airline...who falsified most of his time. In fact, one day I happened into an FBO and found a picture of him on a wall as a solo student...taken only a year before he got his job as the assistant chief pilot for a particular operation. He was a con artist. I have no idea where he is today, but I also have no illusions about the fact that many pilots do the same thing.
When I completed my ATP checkride, the examiner told me that the next step would be to falsify a lot of multi time and go get that first multi job. This was an examiner, mind you. He then went on to say that if I had a conscience crisis about it, I could always simply not log time later on. I elected to ignore his counsel.
Not long ago I was asked to fly with a young man as an evaluation, prior to his Part 135 checkride with the FAA. Before we ever made it to the airplane, numerous red flags were raised. The flight confirmed what I thought. He was lying about his experience. I examined his resume and found inconsistencies, then interviewed him and found more. He began changing his story. Contact was made with former "employers" who had never heard of him, and soon he changed the names on the resume to "downgrade" his experience to something we might not check. We did. He was lying.
I've always maintained that I will know most of what I need to know about an aviator before we ever make it to the airplane. The flight only confirms it.
You'll find more than a few who foolishly falisfy their log, and you'll find more than a few operators too slow or too dense to catch it. A descent one will, because it shows. You can log hours, but can't invent or falsify experience. Hours are superficial, experience is not.
Nobody can tell you what to do. If your signature is in his logbook, however, you are in a position of being dragged into his mess if he screws up. I've been there...my signature was the last one before a pilot stacked up an airplane, and the owners, attorneys and the rest of the free world came to camp on my door step. In my case I had very good records which showed I had no culpability in the matter and had not failed to train or inform the student (a certificated pilot), and the dogs at the door moved on. In your case, who knows? Wouldn't you be better off washing your hands of the lying, cheating, falsifying student by either going to him directly and requesting he destroy the book, or taking it to the next level and reporting him?
Bear in mind that while the noble thing may be going directly to the student, this is not a requirement in your position. If the student has knowingly, intentionally, and willfully falsified his logbook, there's nothing inadvertent here. It's not like you're tactfully whispering to the student that he's made a mistake. He's committed a crime, and one of the building blocks to getting there is your signature; he's invoking your good name in his crime. If he later fails and hurts someone or damages property, or takes a life, now your good name is part of his disaster. Do you really want that?