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Found an old friends laptop that was water damaged a while ago. It turns on when connected to power although the screen doesn’t light up. When connected to another monitor- the screen works. Would replacing the logic board (and battery) be enough to fix the laptop?

@imacow. It is most likely that you do have a logic board issue. Since this is a water damaged MBA you will have residue etc. on the board and most likely shorted out either the display connector or the video circuitry. This would all be fixed by replacing the board. Isolated LCD damage with a fully functioning logicboard would be extremely rare for a water damaged device. Use these guides MacBook Air 13" Mid 2013 Repair to work on your MBA. Remove the logic board and clean everything with 90%+ isopropyl alcohol. Check for damaged components. Post some images of your board with your question so that we can see what you see. Use this guide Adding images to an existing question for that.

If it turns on and works when connected running it through a second monitor, it sounds like the logic board isn’t the part that’s damaged. When you shine a light at the internal display, do you see anything being displayed, or is it completely black? It sounds like just the display is dead, which isn’t as bad as the logic board. It might be a good idea anyway to remove the logic board and look over it to make sure there isn’t any corrosion or other liquid damage on it before you use the computer more. Do you know how bad the liquid damage originally was or what happened to it?