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.
"record of observations, readings, etc.," originally "record of a ship's progress," 1842, sailor's shortening of log-book (1670s), the daily record of a ship's speed, progress, etc., which is from log (n.1) "piece of wood." The book so called because it recorded the speed measurements made by means of a weighted chip of a tree log on the end of a reeled log line (typically 150 to 200 fathoms). The log lay dead in the water, and sailors counted the time it took the line to play out. The line was marked by different numbers of knots, or colored rags, tied at regular intervals; hence the nautical measurement sense of knot (n.). Similar uses of the cognate word are continental Germanic and Scandinavian (such as German Log). General sense "any record of facts entered in order" is by 1913.
It [the log-book] is a journal of all important items happening on shipboard, contains the data from which the navigator determines his position by dead-reckoning ... and is, when properly kept, a complete meteorological journal. On board merchant ships the log is kept by the first officer: on board men-of-war, by the navigator. [Century Dictionary, 1897]