Yes, old log can be delete periodically only if you set from the Magento backend configuration. Store ⟶ Configuration ⟶ Magediary ⟶ Command Runner ⟶ Log ⟶ Auto Clean
No, no one can delete the output log from the Magento backend System ⟶ Command Runner by Magediary ⟶ Log until auto clean set from the backend configuration Store ⟶ Configuration ⟶ Magediary ⟶ Command Runner ⟶ Log ⟶ Auto Clean.