Investigate perl's -i option in perlrun to edit files in place. You'll need to open your log file, too.

Your main loop could look something like:

while ( <$IN> ) {
    print $LOG $_
        if ( s/t/tt/ig );
    print $OUT $_;
}
[download]

A global substitute, as above, isn't going to cause an infinite loop. It'll change

I would like to substitute globally

to

I would like tto substtittutte globally

In-place editing isn't always appropriate, so you may have to write a temp file and rename it once you've processed the input.

Why don't you have a go and come back here if you get stuck?

Good luck!