C:\Windows\Logs\CBS

CBS.log and CBS.persist... files are created (at least) when the System File Checker is run (the usual command is sfc /scannow).  If SFC reports that if has found corrupt files but can't fix them, you can examine CBS.log to determine what those files were.

When CBS.log gets to about 50 MB, the system is supposed to compress it into a CBS.persist file and start a new CBS.log, but it seems that compression operation can fail if the logs are big (> 2GB), if there isn't enough free space, etc.

The usual advice is to keep at least the most recent CBS.log file, but really, it's only useful for troubleshooting.

I'm a bit surprised that you're able to delete them so easily.  You should not be able to do that.  Normally, if you attempt to delete C:\Windows\Logs\CBS\CBS.log you'll get the error message, "The action can't be completed because the file is open in Windows Modules Installer."  The CBS.persist files are a bit easier to delete, but you may have to give administrator permission in order to do so.

Thank You Lem for your quick response.

 Please don't tell my system that I'm not suppose to be able to delete these files!

  I was wondering that the program 'Piniform CCleaner' that will erase all selected temp files would get to the CBS files and then take forever to erase them. Then when I traced the CBS file I saw the enormity of the files. 

 I have not disabled any file reporting of the system. So I don't understand that once I traced down the file itself that the system does not create any more files/reports in it.

 Has Windows started a whole new file that I can't find?

 Available space is not a problem since I have almost 600 gig of free C: HD space.

 C: has only OS and protection programs. All data, to save, storage is kept on a different partition.

 For What They/CBS logs are? question. 

 I have found out they are only created when I run Windows Update.

At least for me.