No. MAVIS doesn't support Log colour space because it's not really possible to have Log on the iPhone. iOS provides all third party apps with a video stream that has REC-709 or REC-2020 already applied - so the range has already been expanded out. Any Log mode we could implement would really just be a filter and could therefore result is lower quality footage. We would recommend keeping your footage as raw as possible and then making any colour corrections required using a colouring package.
Log is, however, definitely something we would like to bring and we are always investigating the possibilities.