1. use the three-argument form of open
2. use lexical filehandles, not globals (you don't know what other code is playing with global globs named L)
3. display the system's error message (from $!)

open(my $log_handle, '>>', $filename)
  or die "can't open log file \"$filename\" for appending: $!";
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As in:

#! /usr/bin/perl
use strict;
use warnings;

my $filename = "logfile.log";
my $string = scalar localtime;

open(my $log_handle, '>>', $filename)
  or die "can't open log file \"$filename\" for appending: $!";

print $log_handle $string;

close $log_handle; # or let $log_handle fall out of scope
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