Is my problem the Logic Board?

Hi! Has the laptop suffered any liquid exposure, or physical accidents?

I would remove the battery and reset the PRAM and PMU, and try things such as removing a memory module and then trying the remaining module in one slot and then the other in order to test the slots. I would try a different AC adapter in case that is the problem, etc. When you have a black screen, if you shine a flashlight through the Apple symbol on the back of the screen do you see an image, or is it completely black?

Sounds like the board, but it could also be a power-related issue caused by a bad DC-in. Is the computer reliable in target mode, or option mode, or if you boot from OS media? It would be useful to know if booting from a CD improves its track record, or if it blanks out doing that as well.