log (n.2) "record of observations, readings, etc.," 1842, sailor's shortening of log-book "daily record of a ship's speed, progress, etc." (1670s), from log (n.1). The book so called because a wooden float at the end of a line was cast out to measure a ship's speed. General sense by 1913.*
So it does have a relationship to the wooden log - it was originally a series of measurements from a floating bit of wood.