Dog it and log it is a phrase in similar vain to such phrases as "bag it and tag it" or "hump it and dump it". In this case, a guy gets a dirty dog out with a chick, typically of a drunken nature, and subsequently records the details of said dogging in a log of some kind.
Bill took a seat in front of the fireplace of his personal library. After putting his glasses on, he logged the dirty details of the previous night's debaucherous encounter.
Bro 1: Yo bro, did you get the dog out with that whore Stacy last night?
Bro 2:Yea bro, I dogged it and logged it.
Both: Dog 'N Log!
Having sex while the female defacates
Gave my wife a dog n' log last night