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Did you install the PBE client using this file: http://l3cdn.riotgames.com/PBE/LoLPBEInstaller2.pkg
Did you install it using this file: http://l3cdn.riotgames.com/PBE/League%20of%20Legends%20PBE.dmg
The former is what you're linked to when you sign up for the PBE, but the latter is what Udyr linked on the forums as the 'updated' PBE installer. I honestly don't know which one is correct, or how they differ once installed.
I've been using the latter, and as such been giving instructions on finding your logs based on this. I suspect the outdated instructions on how to find your logs (http://na.leagueoflegends.com/board/showthread.php?t=3036722) apply to the former, and you'll find your logs (if they exist) in that location.
The logs we're posting aren't created by the LoL game client, because the LoL client never actually runs when it encounters this error, so it can never produce logs. We're providing crash logs from the operating system.
I outlined exact instructions on how to find your log files (both OS level and game logs) with the Live client, and the latter PBE client here: http://na.leagueoflegends.com/board/showthread.php?t=3222596
In any event, the issue has been noticed, and they're working on it: http://na.leagueoflegends.com/board/showthread.php?t=3223344
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Dataslave, thanks for all the info. Regarding how I installed, it was from a post somewhere in the forums and it was the .dmg. No idea why I can't locate the logs you're referring to. I will continue to try to locate them in the various locations given but am fairly certain they simply don't exist (yet).
Thanks again for the all the info. I hope what we've both provided helped Riot solve the issue.