Console.app bounces but does not start
Logs from Console.app are in 3 different places:

/var/log (which is actually /private/var/log)
~/Library/Logs (~ means your home directory)
And /Library/Logs

When looking for general errors, one of those locations should provide insight. Note that Console.app also shows Crash logs, and specifically lists some special log files that may not be in one of the above three directories. However, those are special purpose and won't necessarily provide any insight into the issue at hand.