This is irrelevant but I just have to complain: The new LoggingFilter is really annoying because it forces you to use Java Util Logging. It would be better if it gave me control over the logger. Seems like a step backwards in design.
Since Jersey 2.23, there's a LoggingFeature you could use. The following is a bit simplified example, please note that you can register the feature on WebTarget as well.
All these answers are pretty close but they lack the setting to log the request and response body. At least with Jersey 2.30.1 this is how I accomplish logging the request and response including their respective bodies: