The easiest way to find the product log file is the Help menu, the item name would depend on the IDE version and OS:
Show Log in Explorer
Show Log in Finder
Show Log in Konqueror/Nautilus
Reveal Log in Explorer
Reveal Log in Finder/Finder/Nautilus
Log file is named idea.log and is rotated basing on the file size. When attaching logs to your problem report it's recommended to zip and send several latest log files (idea.log, idea.log.1, idea.log.2, idea.log.3).
Can't access Help menu, how do I find the log file manually?
When IDE doesn't start or you can't access Help menu, locate the log file manually: