How should I Unit test a logging class(file system)?

I have created a very simple log and now I want to unit test it. Here is the

class 

    public class EventLogger : IEventLogger
    {
        private string _logDirectory = @"C:\\EventLog.txt";
        public string LogDirectory
        {
            get
            {
               return _logDirectory;
            }
            set
            {
                if (value.EndsWith(".txt"))
                    _logDirectory = value;
                else
                    throw new ArgumentOutOfRangeException(value, "Log extension was not txt");
            }
        } 

        public void LogEvent(DateTime date, EventCategorie cat,EventType eventType,List<ITrack> tracks)
        {
            if (FS == null)
                FS = File.Open(LogDirectory, FileMode.OpenOrCreate, FileAccess.Write);
            if (!tracks.Any())
                throw new ArgumentException("There was no tracks in the list",nameof(tracks));
            string message = date.ToShortDateString() + " --- " + cat + " --- " + eventType + " --- ";
            foreach (var track in tracks)
                message = message + track.Tag + " & ";

            message = message.Remove(message.Length -3) + Environment.NewLine;
            File.AppendAllText(LogDirectory,message);
        }

I can test on the file extension given and I have done that but I was not sure as of how I could unit test my LogEvent method as I do have a dependency to the file system. I started out googling and most people said that you should use property injection with a filestream. I just felt silly when I startet adding that to the original implementation. I then found this and thought this was one of the cases where you can assume that the static method AppendAllText works, or will I need to create a filestream property just to call this one method So I can test it?