If you want to create your own logout button for a Community Lightning Component, you can use the window.location.replace() javascript function.

The button component (note: this example is the button only, you would typically include this in a Notification):

<aura:component implements="forceCommunity:availableForAllPageTypes" 
 access="global">
    <ui:button label="Logout" press="{!c.logout}"/>
</aura:component>

And in your controller, you can redirect the user to the logout page:

({
    logout : function(component, event, helper) {
        window.location.replace("https://YourCommunityDomain/servlet/networks/switch?startURL=%2Fsecur%2Flogout.jsp");
    }
})

Just replace “YourCommunityDomain” with whatever the base URL is for your community.  Typically something like “MyDomain.na10.force.com”.  This will redirect the user to the standard logout page for your community.