The solution is quite simple. All you have to do is use an 'img' tag between the button tags.
<div class="container">
<div class="dropdown">
<button class="btn btn-primary dropdown-toggle" type="button" data-toggle="dropdown">
<img src="images/example_image.png" alt="dropdown image" class="img-responsive">
<span class="caret"></span>
</button>
<ul class="dropdown-menu">
<li><a href="#">Settings</a></li>
<li><a href="#">Log Out</a></li>
<li><a href="#">Dropdown menu 3</a></li>
</ul>
</div>
</div>
If you still have problems, please attach your code along with you question. Hope this helps.