I checked into the logs, and the second one you posted is actually from a very old version of the game. I can see a crash in that log file there, and it's a known bug that should be fixed in the latest version.
The latest log file is pretty strange, it's just cutting out immediately. If the game is actually crashing out fully, there should be a crash dump folder located in the directory the game is installed to. The folders are labeled with the date of the crash and should have a few misc files in the directory.
You can use Steam to jump to the installation folder for you by right clicking the game in your steam library, selecting properties, Local files tab, and then click "Browse Local Files".
I'd also try doing a complete re-install through steam by choosing the option "Delete Local Game Content", and the re-downloading a clean copy from steam. But, be sure to pull out the crash dump files if there are any first, as clearing the local content would erase them.
Thanks!