Here's an easy way to do it:

#!/usr/bin/perl

use strict;
use warnings;
use diagnostics;

open(LOG, '>', 'LOG_FILE') or die "Can't redirect stdout: $!";
open(CMD, 'ls |');
open(STDERR, '>&', STDOUT) or die "Can't redirect stderr: $!";
open(STDERR, '>', 'LOG_FILE') or die "Can't redirect stderr: $!";
print "LOG_FILE\n";
while (<CMD>) {
    print_stuff ($_);
}

sub print_stuff {
      my ($line) = @_;
      print LOG $line;  
      print $line;
}

close(CMD) or die "close CMD failed: $!";

exit 0;
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