There's no surefire way, I'm afraid to tell you.
However, a damage on the logicboard would mean a damage on the capacitors near the inverter cable, and that's a really, really rare occurence.
The usual troubleshooting for your issue implies:
- Replacing the inverter cable: it usually does the trick if the backlight goes on and off when the display is tilted
- Replacing the inverter, who's usually the culprit for a dim or flickering light
- Replacing the display.
You told that the light worked only in the "low part" of the display. Was it also dim, and affected by a pinkish/yellowish hue, lacking luster?
Your kind of display gets its light source by a CCFL Lamp, something that, for a visual reference, looks like a very thin, stretched neon, lodged in the low area of the display, hidden by a reflective foil that reflects the light evenly.
Thus, despite no one can be absolutely sure that you've not got capacitor issues, I'd try going for a display repair.
Afterall, the CCFL have a shelf life shorter than the rest of the laptop.
I'd telly you to go for a full LCD panel replacement: good refurbished panels are half the price of new ones, and new ones are still less expensive than a logic board.
You could also replace the lamp only, but no one goes that: it's a painstalking work of peeling off the reflective foil, avoiding to crimple it, desoldering the CCFL lamp, soldering back a new one and hope to have done nothing wrong.