Many people rely on server logs to monitor up-time or will ping a server from an in-house location. Unfortunately, this is not adequate as there can be many reasons a website is unavailable.
In shared hosting environments* your bandwidth and availability is potentially at the mercy of dozens of other websites that may or may not be commercial in nature. However, even in dedicated hosting environments outages occur, modems to go off line, routers fail, and networks go down, making it important to monitor availability regardless of your environment.
Your hosting company may provide a monitoring service, but relying on your hosting to monitor availability is a bit like the fox guarding the hen house. In-house monitoring is not adequate, as it will typically will only indicate sever uptime not network availability. I suggest all my clients sign-up for an external monitoring service. Monitoring services check availability from a remote location and when failures occur, they will often ping from alternate locations to ensure it’s not location specific issue.
If you find that you have routine outages you will be have the paper trail to hold your hosting company accountable. Options I have used and like include www.siteuptime.com, www.pingdom.com, and my current favorite www.statuscake.com.
I also recommend you make a point to visit your site periodically during higher traffic periods to see how it is performing. If you see any routine slow downs in your site you need to contact your hosting company for additional processing power or bandwidth (especially true during the holiday months). New visitors to a site will not tolerate delays and will simply click-off to find another resource.
*When managed properly a shared hosting solution can be very reliable. I use shared hosting services for several enthusiast websites of my own and a non-profit where keeping costs low is essential.