How do you log out of the Blio App?
What device are you trying to log out on?
Technically, to log out of the Blio App, you have to deregister the device. (If you go to settings > My Account > Deregister, it will log you out). Please be aware that when you log in again, you will have to re-download all of your books.
To resolve this issue, please uninstall Blio from your iPad and download a fresh copy from the following App Store link.