See my response in
https://news.ycombinator.com/item?id=37877599. "Log" is basically just another transfer function (like srgb, pure-power gamma, etc.). It's confusing because despite the seeming similarities video and still-image people don't use any of the same workflows and thus don't have any shared terminology to talk about things (video people don't use an ICC color managed workflow, and separately think about an EOTF and OOTF. They use the term LUT to describe the transformation a CMS would normally do to convert between color spaces. On the other hand with an ICC-style color management an EOTF and OOTF are by definition inverses of each other).
The main benefit of log as opposed to gamma encoding I can see is that log shares the nice property that each f-stop gets allocated the same number of bits; this results in the fact that while gamma encoding tends to allocate less bits to the brightest parts of an image to allocate things in a "perceptually" fair way (at least according to Steven's power law model), log encoding tries to allocate things in a "physically/optically" fair way, which I guess ends up working out better for editing?
Of course in terms of benefits I suspect that "log" mode also reduces a lot of post-processing.