save output/event log to text file

Hi, I have an rsync ‘clone’ of an external drive working nicely in applescript

 do shell script "sh /SCRIPT/" user name "dan" with administrator privileges

where the shell script reads like this;
rsync -avx --delete-before “/Volumes/A/Shares/” /Volumes/B/Shares/

Can anyone advise how best to automatically save the apple script event log to a text file in /users/dan/Documents/logfiles, ideally with the filename increases incrementally e.g. log1.txt, log2.txt rather than overwrite or add to, one file called log1.txt.