You are opening the file for append (">>") which is write only which means that you can't read from the LOG filehandle.

It would probably be better to use a tied hash instead.

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

my $file_found = 0;
my $desired_filename = 'test1.txt';

# Read log file and look to see if desired file is present
my $logfile = 'file.log';
open my $log_fh, '<', $logfile or die "Can not open $logfile for readi
+ng: $!\n";
while (<$log_fh>) {
    chomp;
    if ($_ eq $desired_filename) {
        $file_found = 1;
        last;
    }
}
close $log_fh or die "Can not close $logfile: $!\n";

# Print result
if ($file_found) {
    print "File $desired_filename found in log file.\n";
}
else {
    print "File $desired_filename NOT found in log file.\n";
}
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$!/usr/bin/perl

use strict;
use warnings;
use List::Util qw(first);

open(LOG, "LogFilename") or die "Unable to open logfile:$!\n";
while(<LOG>){
        print if/\btest1.txt\b/i;
}
close(LOG);
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updated: Fixed typo noted by jwkrahn

I got this right out of Perl for System Administration: Chapter 9: Log Files:

open(LOG, "$LogFilename") or die "Unable to open logfile:$!\n";
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The example from that web page uses open(LOG,"logfile") which is a string literal so the quotes are required. Your example uses a scalar variable so the quotes are not required or desirable, see: What's wrong with always quoting "$vars"?