For information about seeing log messages, look at the answers to How can I view and examine the Android log? Note that the behaviour is different between 4.1-jelly-bean and prior versions.
Android users don't work like users on a normal GNU/Linux system, so unlocking the lock screen is not a "login" in the same way that logging into a terminal session, through an X display manager, or via the network are on a GNU/Linux box. Failed attempts to unlock the lock screen are not logged.
The log is meant as a debugging aid, and log messages aren't stored forever, so the lock screen application would have its own way of storing failed attempts. In general, the Android engineers at Google like to keep the logs quiet on production devices, so system apps have a lot of logging that's turned off in the official build. There's nothing to stop a device OEM, or an unofficial ROM, having more logging for failed unlocks (or any other events). You could compile your own ROM to achieve this, but doing so is way beyond the scope of this site.