The logs are in the /var/log/ directory of the hard drive. Don't mix this up with the /var/log/ directory of the live system! If you've mounted /var/log/ to /mnt, for example, the logs are in /mnt/var/log.
You can also chroot yourself to the /mnt directory:
Then you can run commands as if you were on the original system, e.g.
to find out if there are errors during the boot process.