Unit Test case to check if localstorage is empty once I logout of the Web app
If you are using any I/O (in this case, local storage), you must have fallen into an integration test. 


A unit test could be that when logout is executed, the call to
clearing the local storage is made (using an spy on localStorage). 
An integration test could be that you have data in the local storage and after you have logged out, you don't.