Proof of concept that only works with the exact nesting and file names you specified:

use Archive::Tar;
use IO::Uncompress::Gunzip;
use IO::String;

# Do whatever you want with each file and its data
sub handle {
    my ($name,$data) = @_;
    print "file $name length ", length($data), $/;
}

my $filename = 'a.tgz';
my $outer = Archive::Tar->new($filename);
for my $outerfile ($outer->get_files) {
    my $outerdata = $outer->get_content($outerfile->name);
    my $inner = Archive::Tar->new(
                    IO::Uncompress::Gunzip->new(
                    IO::String->new($outerdata)));
    for my $file ($inner->get_files) {
        next unless $file->name =~ /^.._log$/;
        handle $file->name, $inner->get_content($file->name);
    };
}
[download]

It works, but will likely be dog slow. Since your archive is only 40M maybe it doesn't matter.