Injected ILogger using Autofac doesn't log anything on AzureFunctionTools window (AzureFunctionsV2)

Question

I'm using ILogger abstraction of Microsoft.Extensions.Logging in other to do some logging.

If I do the logging in the Run of the Azure Function it logs to the AzureFunctionTools window without any issues.

But I need logs also in other classes of the application, not only on azure functions, and I want to use the same system!

I'm using Autofac as DI as been described on this answer So after reading a while I did this:

public class AutoFacServiceProviderBuilder : IServiceProviderBuilder
{
    private readonly IConfiguration configuration;

    public AutoFacServiceProviderBuilder(IConfiguration configuration) 
        => this.configuration = configuration;

    public IServiceProvider Build()
    {
        var services = new ServiceCollection();
        services.AddTransient<ITransientService, TransientService>();
        services.AddScoped<IScopedService, ScopedService>();

        var builder = new ContainerBuilder();

        // Registered the ILoggerFactory to provide the logger
        // on the classes where is logging a need 
        builder.Register(s => services.AddLogging().BuildServiceProvider()
            .GetService<ILoggerFactory>()).As<ILoggerFactory>();

        builder.RegisterType<SingletonService>().As<ISingletonService>().SingleInstance(); 

        builder.Populate(services);
        return new AutofacServiceProvider(builder.Build());
    }
}

Then here I've used the logging on a class different than AzureFunction:

public class TransientService : ITransientService
{
    private readonly ILogger logger;
    private int counter = 0;

    public TransientService(ILoggerFactory loggerFactory)
    {
        // I've used this logger name because I've read this (written in 27/May/2018):
        // here: https://www.neovolve.com/2018/04/05/dependency-injection-and-ilogger-in-azure-functions/
        // Azure Functions(v2 beta) applies a filter out of the box for ILogger.The filter looks
        // at whether the logger name is either Function.Something or Function.Something.User.Any 
        // logger that does not have its name matching this format will have its log messages filtered out.
        string loggerName = "Function." + typeof(LazyExample).FullName + ".User";
        this.logger = loggerFactory.CreateLogger(loggerName);
        this.logger.LogError($"I'm creating the class: {nameof(LazyExample)}");
    }

    public int GetCounter()
    {
        this.logger.LogInformation($"Counting {this.counter} on {nameof(TransientService)} class.");
        return this.counter++;
    }
}

In the following image you can see the logs that were made on the function or the internal loggins of azure functions, but my loggings doesn't show:

I've also changed my host.json but my loggings on my service still doesn't log:

{
  "version": "2.0",
  "logging": {
    "fileLoggingMode": "debugOnly",
    "logLevel": {
      "default": "Trace"
    },
    "applicationInsights": {
      "samplingSettings": {
        "isEnabled": true,
        "maxTelemetryItemsPerSecond": 5
      }
    }
  }
}

How can I reuse Logger implementation provided by Microsoft on my own classes and injected by Autofac?