The mail logs are:

/var/log/exim_mainlog -> this is the main exim log
/var/log/exim_rejectlog -> this one only has the rejected mail information
/var/log/maillog -> this one has the mail log in/out info

The way I search for a missing e-mail is:

Log into shell and run:
grep "the@email.address" /var/log/exim_mainlog

For example if you are searching for all info relating to you could run:
grep "" /var/log/exim_mainlog

This will return all the mail activity which contains the email address or domain that you entered. Unfortunately this will not reveal the fail into, you now need to repeat the grep with the message's id to get the entire message process including the fail.

grep "messageID" /var/log/exim_mainlog

For example if the message ID is 1HEw1m-0000qd-OT you would do:
grep "1HEw1m-0000qd-OT" /var/log/exim_mainlog

Next you need to interpret the fail message, some are to the point while others are vague.

Hope this helps. This is how I do it, there may be more efficient methods out there.

(HC) "Where the Kool Kats Host"