how to document.getElementById().innerHTML from different html/php file

Hello I am really new to coding in general but I got the basics of what I need.

I have my index.html that contains this:

<!-- Navigation -->
<nav class="navbar navbar-default navbar-fixed-top">
    <div class="container">

        <!-- Collect the nav links, forms, and other content for toggling -->
        <div class="collapse navbar-collapse" id="bs-example-navbar-collapse-1">
            <ul class="nav navbar-nav navbar-right">
                <li class="hidden">
                    <a href="#page-top"></a>
                </li>
                <li class="page-scroll">
                    <a href="#portfolio">Portfolio</a>
                </li>
                <li id="navbutone" class="page-scroll">
                    <a href="login.php">Login</a>
                </li>
                <li id="navbuttwo" class="page-scroll">
                    <a href="register.php">Register</a>
                </li>
            </ul>
        </div>
        <!-- /.navbar-collapse -->
    </div>
    <!-- /.container-fluid -->
</nav>

keep in mind I got this from a website template that I am editing so I didn't come up with this layout

and I have a php file that has some html in it to try and replace the contents of the list when this part of the code is run:

<?php
if($login_ok) 
    { 
?>
<script type="text/javascript">
function logedin() {
    document.getElementById("one").innerHTML = "<a href="logout.php">Logout</a>";
}
</script>
<script type="text/javascript">
    logedin();
</script>
<?php
header("Location: index.html"); 
        die("Redirecting to: private.php"); 
    } 
?>

This doesn't work and I have no idea if this is even close or not. Thanks in advance for the help. also I might add that they link to login.php where they login through an html form at the bottom of the php.

?> 
<h1>Login</h1> 
<form action="login.php" method="post"> 
Username:<br /> 
<input type="text" name="username" value="<?php echo $submitted_username; ?>" /> 
<br /><br /> 
Password:<br /> 
<input type="password" name="password" value="" /> 
<br /><br /> 
<input type="submit" value="Login" /> 
</form> 
<a href="register.php">Register</a>

<script src="index.html"></script>

</html>