You could change your conditional as follows:

if ($entry5 ne $name && $entry5 !~ m/\.log$/) { # ...
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You have this, currently: $entry5 ne '<*.log>', which doesn't really make sense. You clearly are not looking for a file named <*.log> but you are specifying exactly that within single quotes, which prevent the glob operator from doing anything useful. Additionally you don't really need the glob operator, they way you are testing the files.

There's another puzzler in your code as well: What filename would match $name . '.log' but would not match *.log or m/\.log$/? What I'm getting at is that there doesn't seem to be any good reason to test explicitly for $name, since the part of the conditional to the right of the && operator would already catch everything that ends in .log.

One thing also to bring up: You could probably avoid the readdir and while loop entirely by using the glob operator correctly:

if (-e 'abc' && -d _) {
    foreach my $entry (<*.log>) {
        print OUTPUT ":E: $entry5 is  extra file/dir \n";
    }
}
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This may or may not be exactly what you're after, but it's what your code seems to be trying to do.

Thanks Dave for the comment. !~ m/\.log$/ is working. Apparently you bring up a good point where $name.".log" is extra effort while the right condition has been included it.

For the suggestion to avoid reaadir and while loop, since I cannot sure which file would exist to be searched, and I need to print out whose additional files which not in the spec defined. How would I defined the condition in 'foreach' ?

if (-e 'abc' && -d _) {
    foreach my $entry (#other than <*.log>) {
        print OUTPUT ":E: $entry5 is  extra file/dir \n";
    }
}
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use strict;
use warnings;

my $name = 'user_supplied.txt';
my @errs = grep {$_ !~ /\.log$/ and $_ ne $name} glob(".* *");
print do{local $" = "\n"; "@errs\n"};
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To print the non-log files which are not directories within $dh4, something like this could work...
When using readdir(), remember that the names that you get are simple names from that directory and that in order to do a file test on them you have to prepend the directory name.

opendir my $dh4, $a_directory or die "can't open directory $dh4:$!";
while (my $this_file = readdir $dh4)
{
    next if ($this_file =~ /\.log$/);   #skip all log files
    next if (-d "$dh4/$this_file");     #skip all directories
    print "$this_file is an extra file: $dh4/$this_file\n";
}
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