I'm using Malwarebytes Free, and I can't see where it saves any log files.
The detected "PUP.Optional.Legacy AVG Secure Search" in the attached AdwCleaner[S12].txt, is 1 example of Malwarebytes Free not detecting it.
I ran the scan on Malwarebytes after AdwCleaner detected it, and obviously before I then Quarantined it.
I've had similar examples on other PUP threats. Early on, Malwarebytes detected some, but not all the ones that AdwCleaner did.
I've since uninstalled most of the software that I suspected of creating the PUP threats, and so far have had no more detections.