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]
also logrolling, in the legislative vote-trading sense, "mutual aid given in carrying out several schemes or gaining individual ends," 1823, American English, from the notion of neighbors on the frontier joining forces for rolling logs into heaps after the trees have been felled to clear the land (as in phrase you roll my log and I'll roll yours); see log (n.1) + verbal noun from roll (v.). "Sometimes many neighbors were invited to assist, and a merrymaking followed. [Century Dictionary]. In lumbering, in reference to rolling logs into a stream where they bound together and floated down to the mills.
LOG-ROLLING. 1. In the lumber regions of Maine it is customary for men of different logging camps to appoint days for helping each other in rolling the logs to the river, after they are felled and trimmed — this rolling being about the hardest work incident to the business. Thus the men of three or four camps will unite, say on Monday, to roll for camp No. 1, — on Tuesday for camp No. 2, — on Wednesday for camp No. 3, — and so on, through the whole number of camps within convenient distance of each other. [Bartlett]
However the phrase is not attested in any literal sense, only the political sense, until 1848.