use My::Logger;

open(my $log_fh, '>>', 'blah.log')
  or die "Couldn't open log file.";

My::Logger->set_handle($log_fh);

My::Logger->log("Processing...");
...
My::Logger->log("done.");

##</code><code>##

package My::Logger;

my $fh = *STDERR;

sub set_handle {
   my ($class, $new_fh) = @_;
   $fh = $new_fh;
}

sub log {
    my $msg = join('', @_);
    $msg =~ s/\n+\z/\n/;
    print $fh $msg;
}

1;