This should work but needs some locking on the set.

gistfile1.cs

internal static class Container
    {
        private static class PerType<T>
        {
            public static ILog Log;
        }

        public static ILog GetOrAdd<T>()
        {
            var log = PerType<T>.Log;
            if (log == null)
            {
                Console.WriteLine("new log");
                PerType<T>.Log = new Log1();
                log = PerType<T>.Log;
            }
            Console.WriteLine("return log");
            return log;
        }
    }

    public static class LogExtensions
    {
        public static ILog Log<T>(this T type)
        {
            return Container.GetOrAdd<T>();
        }
    }

Yes, you absolutely could set a container. I might name it something different though since it's only purpose is to hold ILog objects.

I think you may have also missed what I updated, which is that the static gateway ALWAYS returns a new object unless it is initialized with a test logger.

I've updated this for more readability here: https://github.com/ferventcoder/this.Log/blob/master/LoggingExtensions.Core/Logging/Log.cs#L44