Bash Script: Comparing two text files

I am wondering how I can do this efficiently. Given two files, say a.txt and b.txt, I want to write a bash script to do the following:

Every line in a.txt containing '*' that does not exist in b.txt will be added at the end of b.txt with a time stamp.

I suggest you to use a third file for results so you don't mess the b.txt file. You could try this:

cp b.txt c.txt 
for line in $(grep '*' a.txt); do
    # for each line found in a.txt
    echo "Found: $line"
    grep -q $line b.txt   # check its presence in b.txt
    if [ $? -ne 0 ]; then
        # if the result of the grep is not equal to 0
        # it means that the line has not been found in b.txt
        # then print the line in a third file with the leading timestamp
        echo "$(date): $line" >> c.txt
    fi
done

It's clear that maybe you should just improve the grep since I don't know hoe the row where you are searching the '*' is composed.