The only user accessible "logging" that happens is called "Freeze Frame Data" that is normally only saved when certain (usually emissions related) trouble codes are set.
The OBDII Freeze Frame is considered one frame of information taken by an on-board data recorder. When a fault occurs, the PCM stores the input data from various sensors so that technicians can determine under what vehicle operating conditions the failure occurred.
So, what you get is a reading from several sensors that took place at a single time. It's not a running log, and it's not all sensors. It's very, very limited, and the data that is recorded changes based on the trouble code that was set.
That said, if you want to log data, you can log anything you want on your A3 with a cable, a laptop, and software called VCDS (aka. VAG-COM).