Different static home page after login

I have a simple but frustrating problem. I would like to show a different static webpage with different content for customers that have logged in. I have set up two different home pages, “Home Public”, and “Home Private” and a redirect plugin to redirect from the static page “Home Public” to the “Home Private” page after login. This works great but I still have one issue. The Menu bar logo link and any “Home” link always returns to the static page “Home Public” even when logged in??

In short, I need a way to change the static page depending on whether the user is logged in or not so they will return to the correct page when they click on the home button?
Is there a simple way to achieve this?

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