Here's a whole host of questions:
Is it a kernel panic? Does your mac go on kernel panics often? (the dark screen in four languages asking you to restart your mac)
Or is it the spinning beachball of death? Does it just happen with log and capture? Do you have stuff plugged into your firewire ports?
Which is the default connection that your computer is set up to, on launching log and capture. Is it a capture card or firewire?
Does final cut just disappear with a 'love note' later on to tell you that it went on vacation? Does final cut keep crashing constantly?
If final cut crashes or hangs constantly, it would mean that something got corrupted in final cut and it doesn't go down very well with your system. In this case, you might want thrash your preference or do a clean reinstallation of final cut (in this preference order).