<?xml version="1.0" encoding="utf-8"?>
<List>
  <Email_Entry></Email_Entry>
  <Email_Entry EmailID="qw@gmail.com" />
  <Email_Entry EmailID="qw@gmail.com" />
  <Email_Entry EmailID="qw@gmail.com" />
  <Email_Entry EmailID="qw@gmail.com" />
  <Email_Entry EmailID="shssh@gmail.com" />
</List>

protected void btnEmailSubscribe_Click(object sender, EventArgs e)
{
    string EmailID = txtEmailSubscribe.Text;
    string FILE_PATH = Server.MapPath("~/Logs/SubscribedEmails.xml");

    XDocument Emailxml = XDocument.Load(FILE_PATH);
    var EmailExist = Emailxml.Descendants("Email_Entry")
        .Any(x => (string)x.Element("EmailID") == txtEmailSubscribe.Text);
    if (EmailExist)
    {
        lblEmail.Text = "EmailID already exist";
    }
    else
    {
        Emailxml.Root.Add(new XElement("Email_Entry",
                   new XAttribute("EmailID", EmailID)
               ));
        Emailxml.Save(FILE_PATH);
        lblEmail.Text = "Subscribed Successfully";
    }
}

var EmailExist = Emailxml.Descendants("Email_Entry")
        .Any(x => x.Attribute("EmailID").Value == txtEmailSubscribe.Text);