The tertius subdomain site appears to be setup correctly:
[host] => tertius.opnutrition.net [class] => IN [ttl] => 30 [type] => A [ip] => 108.162.194.43
At this point I do not know if the log entries are all blocked hackers and spammers etc. so the current issue is that you do not want to have any log entries from the subdomain site logged in the main site’s Security Log. You would prevent that from happening by creating a rewriterule in the main site’s root htaccess file. See this forum topic for the hierarchical structure (parent to child) of multiple sites under a hosting account and how to do what you want to do: http://forum.ait-pro.com/forums/topic/htaccess-files-for-multiple-website-domains/