You can check email logs in the following location.
/var/log/exim_mainlog
/var/log/maillog
Use "grep domainname /var/log/exim_mainlog" for checking a particular domain.
Similarly "grep emailaddress /var/log/exim_mainlog" for particular email address.
tail -f
/var/log/exim_mainlog for viewing the log in realtime.
Email any time or call 01480 26 00 00 014 80 26 00 00