Help diagnose errors on MacBook that will not boot
Not necessarily a logic board issue (but it is possible!).

1) It may be a RAM Issue: easiest way to test is crack it open (if it's the old non-unibody one then that's not too bad) and try running it with either stick removed. If you get the same issue it's onto...

2) An issue with the installation: there's a chance it could be a weird problem at a very low level of the installation, if you've still got the install discs and don't have any data you can't afford to lose try reinstalling. If you're still stuck...

3) Try the hardware check tool: there's a pretty good guide on how to use it here: on the Mac OS X forums. That should show any issues in detail. However...

4) The processor/logic board is done: and either way if it's either then you're down to opening that puppy up and replacing it.