Simple Injector: Register ILogger<T> by using ILoggerFactory.CreateLogger<T>()

Use the following registrations:

container.RegisterInstance<ILoggerFactory>(loggerFactory);
container.RegisterSingleton(typeof(ILogger<>), typeof(Logger<>));

Or, in case you are integrating Simple Injector into a generic host or ASP.NET Core application, make use of the .AddLogging() extension method to even inject a non-generic ILogger into your application components, as demonstrates in this ASP.NET Core Startup class:

public class Startup
{
    ...

    public void ConfigureServices(IServiceCollection services)
    {
        services.AddLogging(); // Adds logging to the framework

        // AddSimpleInjector enables "cross wiring," which means you can let
        // Simple Injector-resolved components to depend on the generic
        // ILogger<T> abstraction.
        services.AddSimpleInjector(container, options =>
        {
            options.AddAspNetCore();
            
            // AddLogger allows Simple Injector-resolved components to depend on 
            // the non-generic Microsoft.Extensions.Logging.ILogger interface.
            // Simple Injector will automatically inject the correct ILogger<T>
            // for you.
            options.AddLogging();
        });
    }

    ...
}

For a full example, see the ASP.NET Core and ASP.NET Core MVC Integration Guide.

Letting application components depend on ILogger instead of ILogger<T>, makes your code simpler, easier to test, and less error prone. If you're using Simple Injector without Service Collection integration (as the previous example showed, you can use the following registration to let Simple Injector ensure the correct Logger<T> is still injected whenever an ILogger is injected:

container.RegisterConditional(
    typeof(ILogger),
    c => typeof(Logger<>).MakeGenericType(c.Consumer.ImplementationType),
    Lifestyle.Singleton,
    _ => true);

This ensures that every application component gets its own Logger<T> instance, where T is the type of the component the logger is injected into. Take the following class for example that depends on ILogger:

public class ComponentA : IService
{
    public ComponentA(ILogger logger) { ... }
}

The above registration will ensure that ComponentA is injected with a Logger<ComponentA>, even though it simply depends on ILogger and not on ILogger<T>.

You can stop reading here if the above suits your needs... or continue reading if you're interested in a more SOLID solution.

A SOLID solution

Instead of letting application components depend on the framework-defined ILogger abstraction, you could also choose to define an application-specific logger abstraction, as prescribed by the Dependency Inversion Principle (DIP).

The DIP states that abstractions should be defined by the application itself—this means you define your own logger abstraction (also see this for an explanation of why you want to do this) and on top of that you build an adapter, much like described here. You can simply derive your generic adapter from the described MicrosoftLoggingAdapter as follows:

public sealed class MicrosoftLoggingAdapter<T> : MicrosoftLoggingAdapter 
{
    public MicrosoftLoggingAdapter(ILoggerFactory factory) 
        : base(factory.CreateLogger<T>()) { }
}

Using this generic adapter, you can configure Simple Injector as follows:

container.RegisterInstance<ILoggerFactory>(factory);

container.RegisterConditional(
    typeof(MyApplication.Abstractions.ILogger),
    c => typeof(MicrosoftLoggingAdapter<>).MakeGenericType(c.Consumer.ImplementationType),
    Lifestyle.Singleton,
    _ => true);

Based on Steven's solution, I post my answer to help anyone else:

    private void RegisterServices()
    {
        Container.Register(ConfigureLogger, Lifestyle.Singleton);            
        Container.Register(typeof(ILogger<>), typeof(LoggingAdapter<>));
    }

    private ILoggerFactory ConfigureLogger()
    {
        LoggerFactory factory = new LoggerFactory();

        var config = new ConfigurationBuilder()
            .AddJsonFile("logging.json")
            .Build();

        //serilog provider configuration
        var log = new LoggerConfiguration()
                 //.ReadFrom.Configuration(config)
                 .WriteTo
                 .RollingFile(ConfigSettings.LogsPath)
                 .CreateLogger();

        factory.AddSerilog(log);

        return factory;
    }

    public class LoggingAdapter<T> : ILogger<T>
    {
        private readonly Microsoft.Extensions.Logging.ILogger adaptee;          

        public LoggingAdapter(ILoggerFactory factory)
        {
            adaptee = factory.CreateLogger<T>();
        }

        public IDisposable BeginScope<TState>(TState state)
        {
            return adaptee.BeginScope(state);
        }

        public bool IsEnabled(LogLevel logLevel)
        {
            return adaptee.IsEnabled(logLevel);
        }

        public void Log<TState>(LogLevel logLevel, EventId eventId, TState state, Exception exception, Func<TState, Exception, string> formatter)
        {
            adaptee.Log(logLevel, eventId, state, exception, formatter);
        }
    }   

As you can see, my solution is using Serilog as a provider for logging in Microsoft.Extensions.Logging.

Hope it helps!