I implement this behavior in next way.
I create LogOut class:
class LogOut {
    void addOnLogoutListener(LogoutListener listener);
    void removeOnLogoutListener(LogoutListener listener);
    void logout();
}
This is a global class.
When user want to logout he should call logout() methods - this method notify every listener.
In your activity add listener that finishes activity. I always add such listener in BaseActivity class to remove code duplication.
Also in App class you can define listener that starts LoginActivity or clear important resources when Logout is happened.