^^ The Log was something Les put together in 1939-40 at the Epiphone shop, as he was friends with the owner. He approached Gibson around 1941 but they weren't interested.
The LP itself was an in-house job, for the most part, from what I've read, based on some initial designs drawing on their acoustic archtops. One of these was called the Ranger, a solid-body non-cutaway guitar with a single P90. Not long after that, they added a florentine cutaway à la ES140 - there's less than a handful of these still floating around.
Other guitars which came about before that probably had some influence, though nobody from Gibson has confirmed this, not that they would now either. The Bigsby Travis guitar from 1948, with its single, florentine cutaway body, probably also had an influence. There's also the story of O. W. Appleton, a player who built his own single cut solid body guitar in 1941 and supposedly approached Gibson with it in 1943, but they blew him off. Gibson only became interested when Fender started making waves with their new solidbody guitar.
Regarding the guitars Les himself used, they were pretty much all modified. If you look at old pics of him and Mary Ford from the 50s, all the guitars have been hacked up in various ways, different pups, Kauffmann vibrolas etc. Certainly, with the first ones which had the factory wrap-under tailpiece, he corrected them with a separate bridge and tailpiece which allowed for palm muting. Example: