I wrote a naive servlet application, where I'm printing to System.out and System.err, and when I use Jetty, I'm not seeing any of this output. According to the URL I've linked to, this should work with the following syntax. Here's what they say, and the Jetty startup command they recommend:
You can use etc/jetty-logging.xml to take all System.out and System.err output (from any source) and route it to a rolling log file. To do so, include etc/jetty-logging.xml on Jetty startup.
java -jar start.jar etc/jetty-logging.xml