I don't know how to actually log. I see a log file in the apache tomcat folder but I don't understand much of it.
You can use android.util.Log;
then you can Log out messages with Log.i / Log.e etc depending on what level you want.
for example to log an error from an Exception.
Of course you can use this anywhere in your code to log things. The advantage in this over System.out.print is that the Logs colour the messages, making them easier to spot in the Trace, and have extra functionality if needed.
With that in mind, you could also test
your servlet in the browser using something like POSTMAN
for chrome, which is an excellent free plugin or REST requests. You could even use Chrome dev tools since they are also an excellent resource. Either way you need to see the response you are getting from your server hosting the servlet, it will tell you a lot about whats happening. E.g. if the response code is 404, 500 etc.