There are several approaches depending on:
- the backup script runs outside your script.
- the backup script finishes when that line is shown inside the log.
- the backup log only has (or will have) one line containing the text.
- others...

One possibility in this case:
Code:
#!/bin/ksh
( tail -f backup.log | while read l; do
echo ".\c"
echo $l | grep "the requested operation was successfully completed" > /dev/null 2>&1
(( ! $? )) && exit 0
done ) && echo "string found, continue..."
# whatever to execute after the match, down here...
At least if you want to check the log file "on the fly"...
Regards.