First, please specify what you mean by "won't turn on." You'd be surprised how many people say this, but actually, the computer turns on just fine, just can't boot.
So, is it:
1. Not turning on AT ALL. As in no power anywhere, no sounds, no chime, no power to the caps lock key.
2. Making noises, but screen doesn't come on at all.
3. Turns on, screen comes on, but cannot load OS or can't get past a part of the booting sequence.
If it doesn't TURN ON AT ALL... then it's your logic board.