Configure syslog server in F5 with an irule to see actual internet IP in syslog server

Are you just using HTTP? If so, you can insert it in an X-Forwarded-For header and have your web server parse for that. Otherwise, you can use an iRule to log the client's ip address and send it to whatever logging facility you want. As long as your F5 box is configured to pass the data to the syslog server, that should be plenty easy.