I think it references the female clitoris. Please comment, we are guessing but thought someone would know... I know cutter meaning fine phat ass, but the log part threw us off..
The expression was "how about no and how about you lick my log cutter."
when you're holding up your phone and making faces at it, as though you are taking a selfie, but you're really taking a picture of the person across from you or the wall or anything else that seems interesting but you don't want to be caught dead taking a picture of.
This action is often made more convincing by wiggling the eyebrows or opening the mouth, to pretend you're trying to get a Snapchat filter to work.