Starts crashing after first install
Welcome to the forums N Suen! Sorry you are having troubles. [:S]
Your log shows that you are running an out of date version of L5. When you write that Logos is crashing "after first install," do you mean that the program won't run at all? Have you tried opening to a blank layout? [Press and hold the command key immediately upon startup. Check appropriate box]. If you are unable to update your app through the normal method, you might need to redownload the software engine. There should be a link in your order history.
I don't see anything recent in the Indexer log, are you sure that indexing has started/finished? If indexing has not finished I would wait till it is done and try starting Logos 5 again.
One point — if Logos 5 crashes save (and repost) the log files before starting Logos again
I'll just add as a NOTE to the above recommendations that there is a fatal error in the Awesomium log but it doesn't appear to be the cause of a crash (there is no crash log either).
N Suen, ensure logging is set up as detailed in http://wiki.logos.com/Diagnostic_Logging#Logos_5_for_OS_X
Using the command key and a blank layout worked. Thank you.