The thought of someone keeping the Captain's Log to themselves is really interesting, and I'd be 100% okay with anyone having their character do that. It could make for good conflict.

Tyr himself wouldn't horde the log, but he does tend to look around a lot as it is. Most of it will probably just be looking at everything and making sure nothing looks like it's prone to fall apart, especially when he realizes there aren't any lifeboats as far as he can tell. (That said, he's from a fantasy series so modern ships are probably beyond him. But he'll try!)