Do any iOS video apps record in log color space?
This announcement article dated January 11, 2017 indicates that there will be an upgrade to the app FiLMiC Pro which may accommodate what you are looking for. Note that the FiLMic Pro App Store web page indicates the latest release is 12/23/2016 - I do not know when the update the article speaks of will be available:


  The forthcoming update will introduce a new log mode that allows
  filmmakers to shoot videos with a flat picture profile containing
  additional stops of dynamic range.


Also check out FiLMic's website - I couldn't find any details, but I was interested to read that they develop it with Apple's Swift language.

Of note, they have apparently released a beta version for testing and demo of the LOG capture. Compare in particular the dark area of the dashboard in the lower left hand corner in this example of source video vs. the same footage after grading with BlackMagic Design's DaVinci Resolve. Disclaimer - these videos are from the article and the quality of the color grading vs youtube compression and such is relative. Also, as an example of the extra detail shooting this way may afford, note in the third shot, the shadow detail of the hat before and after.

Likely you don't need this information, but here is a recent article which provides a decent overview of what it means to shoot in log color space , or, record a "flat color profile." Also, some related VideoSE Q&amp;A about shooting considerations.