I think this one should be perfect for you. 
We need 3 files
- index.php
- login.js
- login.php
That mean when user submit [index.php] script js file [login.js] running ajax process script [json] in login.js by collect all data from form input [index.php] and send and run script login.php ... This is powerful script of ajax & json
check code below
index.php
<!DOCTYPE html>
<html>
<head>
<script src="https://ajax.googleapis.com/ajax/libs/jquery/1.12.0/jquery.min.js"></script>
<script src="login.js" type="text/javascript" charset="utf-8">  </script>
</head>
    <body>      
        <form method="post" id="login" name="login">
            Email: <input type="email" id="log-mail" name="log-mail"  > <br />
            Password: <input type="password" id="log-pass" name="log-pass" > <br />
            <button type="submit" >Log in</button>
        </form>
    </body>
</html>
login.js
$(document).ready(function(){
    // #login = login is the name of our login form
    $('#login').submit(function(e) {
        $.ajax({
            type: "POST",
            url: "login.php",
            data: $('#login').serialize(),
            dataType: "json",
            success: function(msg){
                if(parseInt(msg.status)==1)
                {
                    window.location=msg.txt;
                }
                else if(parseInt(msg.status)==0)
                {
                    window.location=msg.txt;
                }                               
            }
        });
        e.preventDefault();
    });
});
login.php
<?php
    if(isset($_POST['log-mail']) && $_POST['log-mail'] != ''  && isset($_POST['log-pass']) && $_POST['log-pass'] != '' ) {  
        $_data_ = 'index.php?user_data='.$_POST['log-mail'];
        echo msg_result(1,$_data_);
    }else{
        $msg_att = "index.php?login_attempt=1";
        echo msg_result(0,$msg_att);   
    }
    function msg_result($status,$txt) {
        return '{"status":'.$status.',"txt":"'.$txt.'"}';
    }
?>
you can see on your url if you 
- complete all field => ...index.php?user_data=user_data@gmail.com
- uncomplete => ...index.php?login_attempt=1
Hope this solve your issue