After not touching a couple of Drupal sites for a while I wanted to check what modules needed to be updated. It was at this point that any attempt to get into the administration section produced a blank screen. After looking at the error_log file in the Drupal root directory on the server it indicated that PHP was unable to allocate memory. A Google showed the quickest solution was to add ini_set('memory_limit', '24M'); to the /sites/default/setting.php file. This got the sites working again but it does leaves the question of how soon before that limit is also reached.