Can I get new logic board without serial number? Machine was refurb and so serial on base is wrong

My MacBook is mid 2010 dual 2.4gz unibody white. I bought it refurbished from a company (not apple). It no longer turns on and showed no sign of degradation so I took it to the Genius Bar and they reckon its the logic board.

Problem is that the base on the laptop was replaced before I got it and so the serial number is not the right one and as it doesn't turn on, there is no way of finding it out.

I am wondering if it is going to be fixable?

Thanks in advance.