You can make use of Enterprise Splunk to do the centralized log monitoring of your system as well as application components. For personal use free download provides indexing of up to 500MB per day which is very good limit.
I am using it in Ubuntu 14.04 LTS Desktop as well on another laptop(running Ubuntu 15.10 Desktop) for doing the centralized log monitoring, debugging and visualization.
You can download it from here
I have installed it under /opt
and it runs on the default port 8000.
I am using splunk to monitor all the log files under /var/log
and it looks like this. I can do a free text search on any of the error or string in the log files and it works great.
You can monitor multiple directories also.
Using splunk you can avoid doing a lot of shell scripting and pattern matching and good part everything is provided out of the box.
There are other tools like ELK(ElasticSearch Logstash Kibana) as mentioned here and SiLK.
Hope this helps.