When $base_url is set to http://example.com, the https is rewriten to http.

When $base_url is set in the settings.php to http://example.com, the https is rewritten to http and the form is submitted over clear text. When I look at the source of the login page, the form action is correct:

But when I look at my webserver logs, I don't see any https requests. I simply see the http.

If I don't set $base_url, the login page is submitted using https but the whole site navigation becomes https as well. I was hoping that once the form information was submitted via https that the site could be navigated using http.