logged adj.

unshaped large piece of tree, early 14c., of unknown origin. Old Norse had lag "felled tree" (from stem of liggja "to lie," hence "a tree that lies prostrate"), but many etymologists deny on phonological grounds that this can be the root of English log. Instead, they suggest an independent formation meant to "express the notion of something massive by a word of appropriate sound" [OED, which compares clog (n.) in its original Middle English sense "lump of wood"].

Log cabin (1770) was the typical dwelling of the poor in antebellum U.S. history in the well-timbered region that was then the West. It has been a figure of the honest pioneer since the 1840 presidential campaign of William Henry Harrison (the original application was derisive and either way it was inaccurate). Falling off a log as a type of something easy to do is from 1839.

1759 (in an account of the Battle of Lagos in "Universal Magazine," September), from water (n.1) + log (n.1); the notion apparently is of "reduce to a log-like condition." Compare logged.

WATER LOGGED, the state of a ship when, by receiving a great quantity of water into her hold, by leaking, &c., she has become heavy and inactive upon the sea, so as to yield without resistance to the efforts of every wave rushing over her decks. As, in this dangerous situation, the center of gravity is no longer fixed, but fluctuating from place to place, the stability of the ship is utterly lost. She is therefore almost totally deprived of the use of her sails, which would operate to overset her, or press the head under water. Hence there is no resource for the crew, except to free her by the pumps, or to abandon her by the boats as soon as possible. [William Falconer, "An Universal Dictionary of the Marine," London, 1784]

The verb waterlog (1779) appears to be a back-formation.