Yes. Here's the heart of the answer:
Save Text File( "$desktop/log.txt",
Concat Items( Get Log(), "\!N" ),
Mode( If( File Exists("$desktop/log.txt"), "append", "replace" ) ) )
As shown, it adds some unwanted text to the log, and may not add some text you do want. If you make it an add-in, nothing is added to the log.
You might want a separator, or a date. You could concatenate those to the Concat Items() result.
You might want a file name prompt; you can add that as well.
Edit: And it might make sense to Clear Log() if you plan to do multiple appends in one session.
Craige