This little hack enables fatal error logging for your site, without creating an error log that is insanely big.

logging-helper.php

<?php
/**
 * This changes logging to only log fatal errors. This file should go in your mu-plugins directory.
 */

// Set the error logging to only log fatal errors
error_reporting( E_ERROR );

// Optional: change the location of your error log, it might be wise to put it outside your WP content dir.
// If you don't change it, the default place for this log is debug.log in your WP_CONTENT_DIR.
ini_set( 'error_log', WP_CONTENT_DIR . '/fatal-error.log' );

// Disable Akismets awfully verbose logging
add_filter( 'akismet_debug_log', '__return_false' );

wp-config.php

<?php
/**
 * These three lines should go in your wp-config.php 
 */
define( 'WP_DEBUG', true );
define( 'WP_DEBUG_LOG', true );
define( 'WP_DEBUG_DISPLAY', false );