log (n.1) Look up log at Dictionary.com
unshaped large piece of tree, early 14c., of unknown origin. Old Norse had lag "felled tree" (from stem of liggja "to lie"), but on phonological grounds many etymologists deny that this is 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 compares clog (n.) in its original Middle English sense "lump of wood." Log cabin (1770) in American English has been a figure of the honest pioneer since the 1840 presidential campaign of William Henry Harrison. Falling off a log as a type of something easy to do is from 1839.
log (v.2) Look up log at Dictionary.com
"to enter into a log-book," 1823, from log (n.2). Meaning "to attain (a speed) as noted in a log" is recorded by 1883. Related: Logged; logging.
log (n.2) Look up log at Dictionary.com
"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) which is 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.
log (v.1) Look up log at Dictionary.com
"to fell a tree," 1717; earlier "to strip a tree" (1690s), from log (n.1). Related: Logged; logging.