Be careful, the above code is NOT thread safe. If you have multiple threads using the same logger you'll end up crashing said thread due to unhandled exception in the exception handler or simply corrupting your log file.
It's good for single threaded applications, though.
Edit: Having a second look I'd vote against this method in general. If you know you will dump everything in a file no matter what it's better to write to that file and read it back all up if you want to display it in the UI. This way you are simply wasting resources, especially if your log file grows large.
Also, appending to a memo has it's performance penalty which is extremely painful if items are arriving rapidly.