Is there a general Logging service on Azure?

Azure Monitor is the service you're looking for.

If your apps are already instrumented using Application Insights, you can send custom events and metrics:

https://docs.microsoft.com/en-us/azure/azure-monitor/app/api-custom-events-metrics

and lastly, you can query using Kusto queries:

https://docs.microsoft.com/en-us/azure/azure-monitor/log-query/log-query-overview

A single Application Insights instance can be fed from different applications. You just have to use the same InstrumentationKey. The ApplicationInsight API allow this.

The built in reports will then not be of much use to you. You'll probably have to set up your own reports. But if you do that in a clever way and log custom properties for seperating the different applications I think you can build a great logging solution.

https://docs.microsoft.com/en-us/azure/azure-monitor/app/api-custom-events-metrics

There is N number of services available depending on the requirements or the things you want to analyze, mainly being the Azure Monitor and Application Insights.

Here is the diagram which lists down the available services for the full stack monitoring on Applications and Infrastructure

For the network monitoring, you can use the Network watcher as well.