You needn't pull the whole file into memory just to search for one line.
The caret character,(^), means start of the line in a regex. Read perlre, these meta-characters is the second subject covered.
You can match save and exit the loop in one statement.

sub get_file_from_log(){
   my $log_name=shift;
   open(my $log_file, "<", $log_name);
   my $file_name;
   while (<logFile>){
      last if ($file_name) = /^Generated filename\s*=\s*(.+)$/;
   }
   return $file_name;
}
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