private static void RegisterType(ContainerBuilder builder)
{
    builder.RegisterType<MyLogger>().As<ILogger>();  //MyLogger принимает в конструкторе IWriter
    builder.RegisterType<SimpleWriter>().As<IWriter>();
    builder.RegisterType<LogForm>();
}

public LogForm(ILogger logger)
{
    Logger = logger;
    InitializeComponent();
}

LogForm можно создать через резолв контейнера (т.к. весь граф ее зависимостей зарегистрирован).
Как лучше создать LogForm?

private void btn_OpenForm_Click(object sender, EventArgs e)
{
    using (var scope = AutofacConfig.Container.BeginLifetimeScope())
    {
       var logForm = scope.Resolve<LogForm>();
       logForm.Show();
    }
}

private void btn_OpenForm_Click(object sender, EventArgs e)
{
  var logForm = AutofacConfig.Container.Resolve<LogForm>();
  logForm.Show();
}

private readonly ILifetimeScope _lfScope;
private void btn_OpenForm_Click(object sender, EventArgs e)
{
     _lfScope = AutofacConfig.Container.BeginLifetimeScope())
      var logForm = _lfScope.Resolve<LogForm>();
      logForm.Show();
}

private void btn_OpenForm_Click(object sender, EventArgs e)
{
    var logForm = new LogForm();//Создавать без DI и зависимостей.
    logForm.Show();
}

//Изменим класс формы LogForm
public partial class LogForm : Form
{
    public ILogger Logger { get; set; }            //Сервис
    private readonly ILifetimeScope _lfScope;      //Управление временем жизни сервиса

    public LogForm()
    {
     _lfScope = AutofacConfig.Container.BeginLifetimeScope();
     Logger = _lfScope.Resolve<ILogger>();

     InitializeComponent();
    }


    public void DoSomething()
    {
        //............
    }


    protected override void OnClosed(EventArgs e)
    {
        _lfScope.Dispose();
        base.OnClosed(e);
    }
}

builder.RegisterType<NyLogger>().Named<ILogger>("11").InstancePerLifetimeScope();

builder.RegisterType<OtherLogger>().Named<ILogger>("22").InstancePerLifetimeScope();

 builder.RegisterType<LogForm>()
  .WithParameter(new ResolvedParameter(
      (pi, ctx) => pi.ParameterType == typeof(ILogger),
       (pi, ctx) => ctx.ResolveNamed<ILogger>("11")
       )).InstancePerDependency();

public partial class DoubleLogForm : Form
{
    public ILogger Logger1 { get; set; }            //Сервис
    public ILogger Logger2 { get; set; }            //Сервис
    private readonly IDisposable _loggerOwner1;
    private readonly IDisposable _loggerOwner2;

    public DoubleLogForm(Owned<ILogger> logger1, Owned<ILogger> logger2)
    {
        Logger1 = logger1.Value;
        _loggerOwner1 = logger1;

        Logger2 = logger2.Value;
        _loggerOwner2 = logger2;

        InitializeComponent();
    }
 }


    builder.RegisterType<NyLogger>().Named<ILogger>("NyLogger").InstancePerLifetimeScope();
    builder.RegisterType<OtherLogger>().Named<ILogger>("OtherLogger").InstancePerLifetimeScope();

    builder.RegisterType<DoubleLogForm>()
        .WithParameters(new List<ResolvedParameter> {
            new ResolvedParameter(
                (pi, ctx) => pi.ParameterType == typeof(Owned<ILogger>),
                (pi, ctx) => ctx.ResolveNamed<Owned<ILogger>>("NyLogger")),
            new ResolvedParameter(
                (pi, ctx) => pi.ParameterType == typeof(Owned<ILogger>),
                (pi, ctx) => ctx.ResolveNamed<Owned<ILogger>>("OtherLogger"))
        }).InstancePerDependency();

public partial class LogForm : Form
{
    private readonly Owned<ILogger> Logger;

    public LogForm(Owned<ILogger> logger)
    {
        this.Logger = logger;
    }

    protected override void OnClosed(EventArgs e)
    {
        Logger.Dispose();
        base.OnClosed(e);
    }
}

public partial class LogForm : Form
{
    private readonly ILogger Logger;
    private readonly IDisposable LoggerOwner;

    public LogForm(Owned<ILogger> logger)
    {
        this.Logger = logger.Value;
        this.LoggerOwner = logger;
    }

    protected override void OnClosed(EventArgs e)
    {
        LoggerOwner.Dispose();
        base.OnClosed(e);
    }
}

private readonly Func<LogForm> logFormFactory;

public MainForm(Func<LogForm> logFormFactory)
{
    this.logFormFactory = logFormFactory;
}

private void btn_OpenForm_Click(object sender, EventArgs e)
{
    logFormFactory().Show();
}