To centralize the logs from our many servers we want to use logstash. This makes it easy to overview all our customers servers and catch errors in time. In order to forward logs to an external service where logstash is running the client server (you) need to install Logstash forwarder. See this tutorial: https://www.digitalocean.com/community/tutorials/how-to-install-elasticsearch-logstash-and-kibana-4-on-ubuntu-14-04#configure-logstash-forwarder
You would need to add a configuration form to your control panel to set the logstashserverprivate_address variable.