From Birnam Designs Wiki

Exim's log files are typically at:

• /var/log/exim/log/exim_mainlog
• /var/log/exim/log/exim_rejectlog
• /var/log/exim/log/exim_paniclog

alternate locations are at /var/spool/exim/log

To find out, run:

exim -bP log_file_path