You're changing the value of $zips but that doesn't change the name of the file any more than writing "deposit $4_000_000_000" in your checkbook will add money to your checking account; you've got to use rename to tell the operating system to rename the file (alas, I don't know the Perl function to add money to my checking account).

Try something like this:

opendir(DIR,$destinationdirectory) or die "Could not open $destination
+directory because $!\n";

foreach (my $zips = readdir(DIR)){
     ($txt = $zips) =~ s/\.log$/.txt/i;

      unless(rename($zips, $txt)){
        warn "could not rename $zips to $txt because $!\n";
        next;
      }
}
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Warning Code was NOT tested.

---

minor update

In addition to the advice about collisions, you should probably check permissions on the directory holding the files to be renamed.