1. Reach a usable system
Obviously, it will be impossible to acquire those logs if you can’t have access to a usable system.
See: [HowTo] Reach a minimal system
2. Find the faulty application
If you don’t know which application failed – this is usually the case on a system or graphical failure – a good place to search the culprit is in the system journal .
Supposing the failure occurred in the previous boot:
journalctl --boot=-1 --priority=3 --catalog --no-pager