my @filenames = qw(log1.log logX.log logZ.log);

open(my $out, '>', 'output.txt') or die "Cannot open file 'output.txt' for writing: $!";

for (my $filename (@filenames)
{
    open(my $in, '<', $filename) or die "Cannot open file '$filename' for reading: $!";

    while (<$in>)
    {
        ...

            if (/AS \s+ $keyword/ix)
            {
                print $out join('', @block);
                last;
            }

        ...
    }

    close $in or die "Cannot close file '$filename': $!";
}

close $out or die "Cannot close file 'output.txt: $!";