It depends on what happend when the database went down. If it was killed from the outside, there’s no real evidence unless you have a watch-dog that can tell you when it lost contact. OEM does a lot of that for you, and it’s worth your while to have tools like that operational.
If the database closed itself or was stopped by a user, information about the shutdown will be in the alert-file.
However – if someone removed the power from the box, nothing gets logged. Only a watchdog on a separate box will be able to catch such events.