Leave a constant ping run to 8.8.8.8. Your goal is to see if it is only a dns outage or if you are actually losing the internet.
If it is just DNS then you can set the DNS manually in your pc to 8.8.8.8. The default dns setting have the router act as a proxy to the ISP DNS. You can have issue with either the proxy stuff or the ISP DNS. 8.8.8.8 is a google dns server that seldom has any issues. It gets good performance because the IP address is actually duplicated in many cities.
If you lose a lot of ping packets...especially if you lose a bunch in a row you might have a issue with your internet connection. The messages in the modem log don't mean a lot unless they happen a lot. Most times they don't actually mean...