> How would you claim it's a no log VPN?

Mullvad have been taken to court over this in relation to a copyright infringement case.

TL;DR The judge permitted people to take a fine-tooth comb to Mullvad's infrastructure and no logging was found[1].

[1] https://mullvad.net/en/blog/mullvad-vpn-was-subject-to-a-sea...