/var/log/apache2/error.log
[Sun Dec 18 06:25:04.024324 2016] [mpm_prefork:notice] [pid 17291] AH00163: Apache/2.4.18 (Ubuntu) OpenSSL/1.0.2g configured -- resuming normal operations
[Sun Dec 18 06:25:04.024408 2016] [core:notice] [pid 17291] AH00094: Command line: '/usr/sbin/apache2'
/var/log/mysql/error.log
empty
Logs you posted doesn't present problem. Moreover, it's normal - Apache can start itself multiple times to serve requests.
Take a look at ServerFault thread - Apache is OK, but what is this in error.log - [mpm_prefork:notice]?. It explained this log entries pretty well.
Isn't there more things in your error.log? With this part, it's not possible to debug it.
In meanwhile what's your server size? Maybe you can verify that instances doesn't get killed because of Out of memory. Check syslog, /var/log/syslog, and make sure there isn't any OOM, OOM killer entries for Apache. Probably not a problem, but check it.
Nothing more in my error.log
Server : 512 MB Memory / 20 GB Disk / TOR1 - Ubuntu Wordpress 4.7 on 16.04
No OOM entries in my /var/log/syslog
Usually, the connection failed few second after I update a page. Still good for a post.