Would like to render alt tag for logo but not show site name on the page

I am new to Omega and just finding my way around. I noticed that if I toggle site name off in the theme settings, the alt tag for the logo I uploaded for the theme does not display. If I toggle site name on in the theme settings, then the logo indeed has the alt tag but then the site name appears on the page which I do not want. I am not sure if there is some other setting so I can display my logo but not the sitename but still have the site name associated with the logo? I hope this makes sense. I am pretty sure this is $vars['logo_img'] = $vars['logo'] ? '<img src="' . $vars['logo'] . '" alt="' . check_plain($vars['site_name']) . '" id="logo" />' : ''; in template.php