When a script doesn't work as expected, you need to debug it. To aid in that venture, be verbose. Don't get fancy and try to do everything in a single line (eg. my $log_file = (split /\./, $0)[0] . '.log';). Rather, do the minimum possible in each step, and display all intermediate results, such as:

#!/usr/local/bin/perl -w
# my_script_name.pl
use strict;

my $arg0 = $0;
print "Debug>  Before split -- arg0 = '$arg0'\n";

my @words = split /\./, $0;
my $count = @words;
print "Debug>  After split -- $count words:  [@words]\n";

my $base = $words[0];
print "Debug>  First word (word[0]) = '$base'\n";

my $log_file = $base . '.log';
print "Debug>  Full logfile name = '$log_file'\n";
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You should never find yourself saying "I know the contents of the variable should be X at this point". Rather, "I think the contents of the variable should be X, so I'll print them out to be sure..."