Captain's log, this is where I pinch it off

from Captain Murphy – Children of the Atom Lyrics on Rap Genius

Meaning

A logbook, pretty self-explanatory, is a place where a ship or plane would keep track of their travels should their electronic’s fail. So rather than this being the log of a captain of a ship, it is Captain Murphy’s log. The captain’s log was also a notable part of the Star Trek series, as it was frequently used to show viewers key plot points.

Additionally, “pinching off a log” is a silly, but nonetheless popular way to refer to taking a shit. To take it further, “dropping shit” is one way musicians refer to releasing new music.

Finally, considering the sexual theme of the song, he could certainly be referring to his penis, which log is slang for.

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