$ sudo apt-get remove rsyslog 

Uninstall rsyslog including dependent package

If you would like to remove rsyslog and it's dependent packages which are no longer needed from Ubuntu,

$ sudo apt-get remove --auto-remove rsyslog 

Use Purging rsyslog

If you use with purge options to rsyslog package all the configuration and dependent packages will be removed.

$ sudo apt-get purge rsyslog 

If you use purge options along with auto remove, will be removed everything regarding the package, It's really useful when you want to reinstall again.

$ sudo apt-get purge --auto-remove rsyslog 

See Also,