Is $site_name , $logo ,$site_slogan are part of render array $page?
The variables you use in templates aren't really part of the render arrays themselves.  They are created by the various preprocess and process functions and added to the $variables array.  By default, those you list are added in either a template_preprocess_page() or template_preprocess_html(), which your theme can have, too.  Look at the source to see what is going on.  For example, the stock template_preprocess_page sets $site_slogan like

$variables['site_slogan'] = (theme_get_setting('toggle_slogan') ?
                            filter_xss_admin(variable_get('site_slogan', '')) : '');


To aid in figuring this out, I suggest installing the Devel module, and looking at the functions that it provides.  For example, you can do

dpr($variables);


to get a dump of everything in the $variables array, or you can do

kpr($page);


to poke through the $page object via Krumo, which is included in Devel.