NNewcomb wrote on May 28^th, 2012 at 1:46am:

Pinatubo wrote on May 27th, 2012 at 7:50pm: NNewcomb, First you must uninstall FS from the old computer, and then install it on your new one. During registration process, when entering key product, MS knows when you are trying to install FS on more than one PC. Don't forget to backup [Logbook.BIN] file (Start > Documents > Flight Sinulator X Files > Logbook.BIN) to avoid losing your flight hours and awards earned. After installing, paste saved [Logbook.BIN] file on new [Flight Simulator X Files]. Pinatubo. My logbook file was corrupt, and prevented me from opening FSX. SO I had to delete it  :(

I use a logbook repair utility that fixes the logbook, but usually with the loss of only the last few flights. The program is called logbook editor.

Here is a little bit of info from the readme.txt file:
This software is a graphical editor of Flight Simulator X logbook files.
It can modify fields not editable within FSX (including fligt durations)
and repair broken logbooks. It can also export logbook to HTML or
excel-compatibles files.

It is the second last item on this page at Simviation: http://www.simviation.com/fsxmisc1.htm.
I highly recommend it. The version I am using is 0.3 and I'm not sure if there is a later version available.

Mark