And what if it is not successful? Wouldn't you like it to tell you what happened? Or that it failed? (And what if it spelled successful correctly, and your grep didn't find it?
A little creativity with scripting can resolve those questions. Any command returns a success code, and those can be used in IF statements. So, with
if [ 'grep success /tmp/backup/$File.log' ]
you could put appropriate information in your report whether it succeeds or fails.