Remember: GPEX Wallet isn't like a traditional website, where you're logging in to a remote server. Rather, GPEX Wallet stores your Secret Recovery Phrase, private keys, and a certain amount of information about how you use GPEX Wallet (the 'state logs') on your local device, so that whenever you want to carry out an operation on a blockchain, your SRP and private keys are ready to be used.
In this sense, you don't log out of GPEX Wallet, but you can lock the application, so that you'll have to re-enter your password or biometrics to log back in.
Click Lock in the dropdown when clicking on the identicon icon:
You can also set an auto-lock, which you can find in Settings > Advanced:
Here you can set a timer in minutes after which the wallet will be locked.