1398, of unknown origin. O.N. had lag "felled tree" (from stem of liggja "to lie"), but on phonological grounds etymologists deny that this is the root of Eng. log. Instead, they suggest an independent formation meant to "express the notion of something massive by a word of appropriate sound." Logging "act of cutting timber" is from 1706. Logjam "congestion of logs on a river" is from 1885; in the figurative sense it is from 1890. Logrolling in the legislative vote-trading sense first recorded 1823, from the notion of neighbors on the frontier helping one another with the heavy work of clearing land and building cabins (as in phrase you roll my log and I'll roll yours). Log cabin in Amer.Eng. has been a figure of the honest pioneer since the 1840 presidential campaign of William Henry Harrison.
I'm not sure I understand the boldfaced text... From what sense it makes to me, I completely disagree.
I also was wondering if this word might be related to leg.