So you have Exim setup but your logs are looking pretty useless. Maybe you have a spammer on the server and you can’t come close to figuring out the source. Most exim configurations by default have very limited logging enabled and will be useless for any serious sleuthing. Our sample logging configuration should be used on all servers. It ensures you will be able to trace mail and figure out where spam comes from. You should edit /etc/exim.conf and find the log_selector line in the config file. If the line exists replace it with the line below. If it does not then place the line below anywhere in the top section of options. You should then save the file and you can also save a copy in case a control panel or update overwrites it.
log_selector = +address_rewrite +all_parents +arguments +connection_reject +delay_delivery +delivery_size +dnslist_defer +incoming_interface +incoming_port +lost_incoming_connection +queue_run +received_sender +received_recipients +retry_defer +sender_on_delivery +size_reject +skip_delivery +smtp_confirmation +smtp_connection +smtp_protocol_error +smtp_syntax_error +subject +tls_cipher +tls_peerdn
Restart Exim and enjoy your new enhanced logging!