Sounds like your trainer needs some training himself. I used to train for swift long time ago, but never asked a trainee to tear out pages from a logbook. Between the two of you there should be 22 hours of available driving time a day, how many does this guy need? You guys fuel, eat and shower right?
Most trainers train because they can run the truck as a team, but keep all the cash.
I would ask him straight up to please explain why is it necessary to redo the log. You should be able to log right the first time. If he gives you a hard time, you know the rest of your time in his truck is toast, so I would call the company and request a different trainer.
It is best to log it as you do it, especially when you first get out there. If you ever get into a wreck they can go back and find out if you lied on your logs. We all do things we shouldn't to get from A to B, but it is a risk and people get busted all the time.
Good luck and keep us posted