Will replacing the logic board fix it?

Not bad RAM, no RAM

1 beep = no RAM installed

2 beeps = incompatible RAM types

3 beeps = no good banks

4 beeps = no good boot images in the boot ROM (and/or bad sys config block)

5 beeps = processor is not usable

Since the RAM is integrated into the logic board, you are correct the logic board will need to be replaced. Since I don't know your exact model, I can't tell which board that is. After you check prices, you might want to contact Apple and see if you can get a "Flat Rate repair". It usually runs $325 plus tax but may be your best route to repair.