Don't run out and buy a new logic board or video card just yet!! RESET YOUR PRAM!! (This is not the same as SMC reset)
As a potential solution to the black screen on boot, this is done upon restarting the Mac similar to an SMC reset:
Reboot the Mac and as soon as you hear the boot chime, hold down the Command+Option+P+R keys together
When you hear the boot sound again, the PRAM has been reset so let the Mac boot again as usual
At this point your Mac should boot again as usual and no longer have the black display, loading OS X as usual.
If there is no boot chime, don't worry, just hold down the Command+Option+P+R keys together shortly after you hear the typical boot up disc check, or within a couple of seconds of depressing the power button to turn on the Mac.