This pertains to someone with a penis. You can haul a log down the street, haul a log in class (if you happen to get an NRB). Mainly, anytime you are rocking a boner. Log meaning wood, haulin' meaning carrying. Thus, haulin' a log. Carrying a boner. This applies to transvestites, as some may have a penis now.
I can't stand up right now. I'm haulin' a log.
or
I bumped into Carol as I was haulin' a log. She seemed to enjoy it.
or
Rick was haulin' a log this morning when he walked downstairs to get breakfast, which shocked mom. and ruined everyone's morning.