Logout form within Bootstrap navbar dropdown

That will not really work, but you can create the logout-form somewhere else:

<form id="logout-form" style="display:none" action="/logout" method="POST"></form>

and when the user clicks on logout you just call

document.getElementById("logout-form").submit();

in Javascript or you set

onclick="document.getElementById('logout-form').submit();"

try this Sample [UPDATED]

.btn-logout {
  width: 100%;
  padding: 3px 20px !important;
  text-align: left !important;
}
<link href="https://maxcdn.bootstrapcdn.com/bootstrap/3.3.7/css/bootstrap.min.css" rel="stylesheet"/>
<script src="https://ajax.googleapis.com/ajax/libs/jquery/2.1.1/jquery.min.js"></script>
<script src="https://maxcdn.bootstrapcdn.com/bootstrap/3.3.7/js/bootstrap.min.js"></script>
<div class="container-fluid">
  <div class="row">
    <div class="col-lg-12">
      <nav class="navbar navbar-default">
        <div class="container-fluid">
          <!-- Brand and toggle get grouped for better mobile display -->
          <div class="navbar-header">
            <button type="button" class="navbar-toggle collapsed" data-toggle="collapse" data-target="#bs-example-navbar-collapse-1" aria-expanded="false">
              <span class="sr-only">Toggle navigation</span>
              <span class="icon-bar"></span>
              <span class="icon-bar"></span>
              <span class="icon-bar"></span>
            </button>
            <a class="navbar-brand" href="#">Brand</a>
          </div>

          <!-- 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">
              <li class="active"><a href="#">Link <span class="sr-only">(current)</span></a>
              </li>
              <li><a href="#">Link</a>
              </li>
              <li class="dropdown">
                <a href="#" class="dropdown-toggle" data-toggle="dropdown" role="button" aria-haspopup="true" aria-expanded="false">Dropdown <span class="caret"></span></a>
                <ul class="dropdown-menu">
                  <li><a href="#">Action</a>
                  </li>
                  <li><a href="#">Another action</a>
                  </li>
                  <li><a href="#">Something else here</a>
                  </li>
                  <li role="separator" class="divider"></li>
                  <li><a href="#">Separated link</a>
                  </li>
                  <li role="separator" class="divider"></li>
                  <li><a href="#">One more separated link</a>
                  </li>
                </ul>
              </li>
            </ul>
            <form class="navbar-form navbar-left">
              <div class="form-group">
                <input type="text" class="form-control" placeholder="Search">
              </div>
              <button type="submit" class="btn btn-default">Submit</button>
            </form>
            <ul class="nav navbar-nav navbar-right">
              <li><a href="#">Link</a>
              </li>
              <li class="dropdown">
                <a href="#" class="dropdown-toggle" data-toggle="dropdown" role="button" aria-haspopup="true" aria-expanded="false">Dropdown <span class="caret"></span></a>
                <ul class="dropdown-menu">
                  <li><a href="#">Action</a>
                  </li>
                  <li><a href="#">Another action</a>
                  </li>
                  <li><a href="#">Something else here</a>
                  </li>
                  <li role="separator" class="divider"></li>
                  <li><a href="#">Separated link</a>
                  </li>
                  <li>
                    <form>
                      <button type="submit" class="btn btn-link btn-logout">Logout</button>
                    </form>
                  </li>
                </ul>
              </li>
            </ul>
          </div>
          <!-- /.navbar-collapse -->
        </div>
        <!-- /.container-fluid -->
      </nav>
    </div>
  </div>
</div>

I have helped you in some way