Write a message to a log
After discussion in comments your code became much better.

Also if directoryPath and fileName will not be changed after initialization in constructor you can define these fields as readonly:

private readonly string directoryPath = string.Empty;
private readonly string fileName = string.Empty;


This code


using (var logFile = new FileStream(Path.Combine(directoryPath,fileName), FileMode.Append, FileAccess.Write))
{
    using (var logFileWriter = new StreamWriter(logFile))
    {
        logFileWriter.Write(message);
    }
}



can be replaced with

File.AppendAllText(Path.Combine(directoryPath, fileName), message);


Also ArgumentException's constructor takes message as first argument and paramName as second, so you need to call it like this:

throw new ArgumentException("Write some error message...", nameof(logDirectoryPath));