We are going to use a small trick for this! We use Azure Application Insights for both our Azure Function and our AI Foundry models.
That offers a bunch of really cool insights right OOTB on both resources:

We love specially the one for Foundry! But also… wait, what is this?

Developers in Azure are used to this window for debugging their Function Apps.

So that we see…

As we can see, right after the Azure function gets a request, all the logs it generates are streamed to Application Insights and, of course, direcly to us. That is done through some cool sockets internally!