The version numbers in there might be slightly different, but the basic path should be right. You guys have done a great job attaching the launcher logs, but it looks like the problem runs a little deeper than that this time around.
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I'm also having issues with the patcher not launching the game. Yesterday in the afternoon it was working just fine. I uninstalled and it froze at the very end of the process. The uninstallation didn't seem to complete.
I went and tried to run lol.launcher.admin.exe because the failed uninstallation had deleted. lol.launcher.exe. It gave me an error saying that the "earlier version" that I tried to download was causing issues with the uninstallation and self-fixed them. But it still doesn't work.
Now I have a weird folder that's labelled "LeagueOfLegends.NA.04_26_2011" beside the default LoL folder. I hope something is worked out here.
EDIT: Here's the first log you requested. However, the last log for the second log you asked for was created yesterday when I last was able to play.