Simplest way to log all messages from an Azure Event Hub
You could write your own worker process to read the messages off EventHub and store them to blob storage. You do not need to do this real time as messages on EH remain for the set retention days. The client that reads the EH is responsible for managing what messages have been processed by keeping track of the EH message partitionid and offset. There is a C# library that makes this extremely easy and scales really well: https://azure.microsoft.com/en-us/documentation/articles/event-hubs-csharp-ephcs-getstarted/