How can I deal with a missing or corrupt binary ("advanced") log table?
As of CiviCRM 4.7, you can fix this by installing any extension; see CRM-15078.  Schema differences are fixed on extension installation.

If you are on an earlier version of CiviCRM, there's a trick you can do.  Edit the PHP for a page that doesn't normally get used - I like the API Explorer page (CRM/Admin/Page/APIExplorer.php).

Right after public function run() {, I put:

$temp = new CRM_Logging_Schema();
$temp-&gt;fixSchemaDifferences(TRUE);
echo "Binary log schema differences are now fixed.";
die;


Once I get back the "now fixed" message, I remove the code from the page and everything works.