To enable extended logging in exim to trace nobody mails. Try the following trick .
1. Edit /etc/exim.conf
2. On the second line add :
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 \
Make sure all that comes on a single line.
3. Save and exit.
4. Restart Exim.
A little bit luck and you should be able to catch the spammer.