JSFiddle - React, Tailwind, and code Playground
HTML
<link rel="stylesheet" href="http://twitter.github.com/bootstrap/assets/css/bootstrap.css">
<script src="https://raw.github.com/imsky/holder/master/holder.js"></script>
<link rel="stylesheet" href="http://twitter.github.io/bootstrap/assets/css/bootstrap-responsive.css">
<div class="navbar navbar-inverse navbar-fixed-top" >
<div class="navbar-inner">
<div class="container-fluid">
<button type="button" class="btn btn-navbar" data-toggle="collapse" data-target=".nav-collapse">
<span class="icon-bar"></span>
<span class="icon-bar"></span>
<span class="icon-bar"></span>
</button>
<a class="brand" href="#"><img src="http://mkowalczyk.webfabryka.pl/wp-content/themes/Kancelaria/images/logo.png"></a>
<div class="nav-collapse collapse container" >
<ul class="nav">
<li class="active">
<a href="#">Home</a>
</li>
</ul>
</div>
</div>
</div>
</div>
CSS
.navbar .brand {
position: fixed;
overflow: visible;
padding-left: 0;
}