Yes, you can check/view the logs for any domains that are owned by your user (you will not be able to view other users' logs). Your logs are stored in the /home/yourusername/logs directory, where you will find a subdirectory for each of your domains. The logs themselves are stored in the subdirectory named for the domain.
You can view these:
1) by logging into the shell and using standard *nix tools to view them, or ...
2) you can login with FTP/SFTP, navigate to the appropriate directory, and download them to your own computer for inspection and/or further processing.