NEW UPDATE
I tried reloading the sim and it did not lose anything. It also did not fix the problem. I then backed up all my aircraft, add on scenery, etc directories and uninstalled the sim using the disk. This left behind a group of files and directories under the fs9 directory. I deleted the whole directory and all of the files and re-installed.
When I tried to re-set my settings in FS9, they were already there just like before I un-installed. This told me there was something else left somewhere. When I tried the sim again, the same problem occured.
I then remembered that there was another place where the program puts files.
c:\documents and settings\[user name]\application data\microsoft\fs9
In that folder is a file named fs9.cfg along with others. When I opened the file in notepad, I found a parameter "log_file=A++++++++++++++++". I then deleted just the file name and left the "log_file=" then restarted the program.
After a short flight, I closed the program and the logbook worked normally. I then opened the cfg file again and the entry read "log_file=logbook" as it should. The old logbook file itself was intact and none of the prior information was lost.
If this problem occurs, the first thing to try is to open the cfg file and change this setting and re-try. It takes about 2 minutes to do and you don't need to re-install anything.
I did report this problem to Microsoft and they told me that they were able to reproduce the problem. Maybe they will find a fix soon.