Yes it would be easier. If you put something like this in your header:
$user =& JFactory::getUser();
if($user->id == 0)
$logo = 'logo1_url';
else
$logo = 'logo2_url';
Then replaced the url for your logo with $logo, and inserted the proper image url's into the code above, It should work the way you want.
Good luck,
Jim