There is no way to "log out" because we rely on the privacy settings of your phone to protect your information and your account is linked to only the device you installed it on.
Because your information is stored in an encrypted state on the blockchain, and the key for that encryption resides only on your phone, you can't login to the wallet from a computer or another device but it also means you don't have to log out and then log back in when you're ready to use the My Bindle app again. It's always ready for you to use.
Using other devices is why most apps have a login and password, so you can access them from other places. If you wish to access your Bindle wallet on another smartphone or a tablet you would have to use the restore procedure (assuming you have backed up your account first, which is strongly recommended). The restore procedure requires providing a password proving you have control of your email, which will trigger a one time password allowing you to restore your account.
For more information on backing up your Bindle wallet, please see this FAQ article.