This is one of the reasons why it's really bad to use some 3rd-party code without source code. I would advise to avoid it by all means. By the way, the problem has nothing to do with registry (thanks goodness, more and more software products these days don't contaminate system registry anymore, but some do).
Of course, it could be possible that the application in question has provisions to switch off the logging, which is typically done in the *.config file, but 1) I believe you already explored this opportunity and you would not ask this question if you could find the way; 2) you did not provide any relevant information on this application to help with this.
In principle, if there are no other way and if the application in question is a .NET application, you can easily reverse-engineer it, remove logging and recompile it. For this purpose, Reflector or, better, open-source ILSpy can be used. Please see:
http://en.wikipedia.org/wiki/.NET_Reflector[
^],
http://ilspy.net[
^].
—SA