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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.
Yes, old log can be delete periodically only if you set from the Magento backend configuration. Store ⟶ Configuration ⟶ Magediary ⟶ Command Runner ⟶ Log ⟶ Auto Clean