I have snow leopard, Mac mini (2007 ), and logic express( which I don't really use).
Time has obviously moved on , but with my versions, you have to have the serial nos. from the. Install discs etc. or better still, the original discs that were purchased ,
However , it sounds like the op has purchased his off the App Store as a download.
If that is the case, then surely , like buying an album /film/ ringtone , this has already been paid for, and there should be no problem Downloading again.
Hopefully the version of logic is compatible with the laptop;)