Exception handling technique between class library and other projects

I'm little bit confused about handling exceptions within a hierarchy of methods.

Let's say, I've a class Logger in a class library project. Please have a look on the following code-

namespace MyLibrary
{
    public class Logger
    {
        //exposed to other project
        public static void CreateLog(String message)
        {
            try
            {
                WriteInFile(message); //calling the method below
            }
            catch (Exception Ex)
            {
                throw Ex;              
            }
        }

        //private method
        private static void WriteInFile(String message)
        {
            try
            {
                //Writing in a file
            }
            catch (Exception Ex)
            {
                throw Ex;
            }
        }
    }
}

Let's assume, I'm using that library in an ASP.NET MVC project. Code -

namespace MvcProject.Controllers
{
    public class HomeController : Controller
    {
        public ActionResult DoSomething()
        {
            try
            {
                Logger.CreateLog("some text.");
            }
            catch (Exception Ex)
            {
                 //Exception is handled here.
            }

            return View();
        }
    }
}

Hierarchy: DoSomething() -> CreateLog() -> WriteInFile()

All the three methods have try.. catch.. blocks. My questions are-

• Do I actually need try.. catch in CreateLog() and WriteInFile() method?

• If I use that in all the methods, does it have any performance impact?

This is a fictitious example to get the problem explained.

It would be more useful for me if you post an answer with a revised code block that you suggest.

Thank you.