Mysql server doesn’t start on Ubuntu 16.04

I'm using Ubuntu 16.04 and recently my database stopped working (don't know why, maybe after an update). When I tried to reinstall mysql-server-5.7 it returned errors on install so I purged everything and switched to mysql-server-5.6, this version installs but it won't run. This is the output:

mysql.service - LSB: Start and stop the mysql database server daemon
   Loaded: loaded (/etc/init.d/mysql; bad; vendor preset: enabled)
   Active: failed (Result: exit-code) since Čet 2017-02-02 10:00:15 CET; 8s ago
     Docs: man:systemd-sysv-generator(8)
  Process: 9225 ExecStart=/etc/init.d/mysql start (code=exited, status=1/FAILURE)
 Main PID: 12545 (code=exited, status=1/FAILURE)

Feb 02 09:59:44 alen-Aspire-E1-570G systemd[1]: Starting LSB: Start and stop the mysql database server daemon...
Feb 02 09:59:44 alen-Aspire-E1-570G mysql[9225]:  * Starting MySQL database server mysqld
Feb 02 09:59:44 alen-Aspire-E1-570G su[9259]: Successful su for mysql by root
Feb 02 09:59:44 alen-Aspire-E1-570G su[9259]: + ??? root:mysql
Feb 02 09:59:44 alen-Aspire-E1-570G su[9259]: pam_unix(su:session): session opened for user mysql by (uid=0)
Feb 02 10:00:15 alen-Aspire-E1-570G mysql[9225]:    ...fail!
Feb 02 10:00:15 alen-Aspire-E1-570G systemd[1]: mysql.service: Control process exited, code=exited status=1
Feb 02 10:00:15 alen-Aspire-E1-570G systemd[1]: Failed to start LSB: Start and stop the mysql database server daemon.
Feb 02 10:00:15 alen-Aspire-E1-570G systemd[1]: mysql.service: Unit entered failed state.
Feb 02 10:00:15 alen-Aspire-E1-570G systemd[1]: mysql.service: Failed with result 'exit-code'.

I hope someone will be able to help me since I'm out of ideas.
EDIT: Mysql error log: http://pastebin.com/tuMj1AA7