HOWTO: Send your logback to Logstash

I recently started playing around with OpenHAB, which is an excellent home automation application. During my "playing" with the system, I decided to try and run it on a Raspberry Pi 3 Model B. I figured that it's a fast enough computer to handle the load of the OpenHAB. One thing that I've read somewhere was that someone managed to kill 3 SD cards due to extensive logging (completely configurable). So, I thought... What if I could completely bypass writing my logs to local filesystem and go straight to ElasticSearch (via Logstash)?

Howto view and analyze your logs on a web page

Recently I started having some problems with my DD-WRT router. I was having some connection problems and occasional reboots. So, to deal with the problem, I wanted to collect some data from the router in the form of logs. Luckily for me, DD-WRT has a syslogd service which is could send logs to a syslog server on another machine over TCP (or UDP?) connection. Since I know close to nothing in that area, I went to the next best trusted source - Google!