How can I tell if the logic board is bad?

I went to use the 13" white macbook for the first time in about 6 weeks and it started right up. I used it for about 5 minutes and when I went to the menu and turned off bluetooth, at that very instant, the whole thing went dark. It seemed to turn off.

Now when I push the power button I don't hear or see anything except when there is complete silence in the room I can hear a very quiet hum which I believe is the hard drive. A few months back I had to replace the hard drive because it was clicking and couldn't be revived.

I have removed the battery for 5 hours, held down the power button for 5 seconds, replaced the battery, plugged in the power and still I get nothing. The only way I can tell the hard drive is spinning is because when I hold the power button again I can hear it spin down.

Could it be anything but the logic board? How can I tell it is the logic board?
I have been unemployed for almost a year so I am not in a position to spend money at apple on this.

Any help is appreciated.

Thank you.