If you have some problem with exim, and the logs are not enough, edit the /etc/conf.d/exim file, and add the debugging option (-d):
# Command-line options for running exim
EXIM_OPTS="-bd -q15m -d"
Then restart exim:
# /etc/init.d/exim restart
The exim daemon will not be detached from the console, the debug information will be printed to the standard output. Good luck!