Off-machine logging. There are commercial services (we're using Papertrail but there are tons of them), roll-your-own solutions (Elasticsearch-Logstash-Kibana), and simple solutions (syslog).

For an easy and simple solution, spin up a second instance and send logs to it via rsyslog over a private network interface. Most mature frameworks provide a method to send logs over syslog. It's UDP and very lightweight. Another plus: if you are compromised, you have another server with your logs and that server isn't running your vulnerable app.