You should check for few things on why the logs are filling at this rate:
Check if the log level for dispatcher logs is ERROR, not DEBUG or TRACE. TRace can produce a lot of lines of log for a single request. Log level can be changed from httpd.conf or apache2.conf file by changing dispatcher global configuration
Try to analyse what type of requests are filling up the logs whether there are logs of flushing/invalidation of content request or page request etc. Then try to work on that part, you might new an extra pair of dispatcher/publisher if single publisher is serving a lot more than its capacity.