$ cat ad.pl
#!/usr/bin/env perl 
use strict;
use warnings;
use autodie;

unlink ('filewhichdoesnotexist');
print "Everything went fine.\n";
$ ./ad.pl 
Can't unlink('filewhichdoesnotexist'): No such file or directory at ./
+ad.pl line 6
$
[download]

So you can simply redirect STDERR to the logfile of your choosing.