Rec.709 (ITU-R BT.709) is the international standard for HD video. It fixes the red, green and blue primaries, the white point and how brightness is encoded, and almost every phone video, YouTube upload and TV broadcast in standard dynamic range uses it. When people talk about a “Rec.709 LUT”, they mean a conversion that turns camera footage into that normal, viewable colour.
Log footage is the opposite of finished. Cameras with a log profile store a wide range of light in a flat, grey-looking curve so there is room to grade later. It must be converted to Rec.709 (or an HDR standard) before it looks right, which is a correction step, not a look. The log to Rec.709 page covers each camera’s profile.
How long grading takes
For a phone clip headed to social media, a correction and a look can take a minute or two. A short commercial may spend a day or more with a colourist, and feature films are commonly graded over a period of weeks, because every shot is balanced against its neighbours and checked on calibrated displays. Most of that time goes into consistency, not into inventing the look.