Rec. 709 and Log C

Log is a data recording method, not a color space.  Logarithmically encoded data is encoded on specific curves, kinda looking like part of a parmetric curve. Rather than being encoded in a linear fashion.

So uncorrected log files tend to look low contrast and washed out for color until you "normalize" them back to the (more) linear normal viewing curve.

To further help confuse things, Premiere Pro now has an auto- function to tonemap log data to the timeline color space for many log formats. So you don't see the 'log' look even to begin with.

Log encoded media isn't supposed to end up looking "log" ... it's just a way to capture data to file. It's expected to be 'normalized' before exporting. Which in the past has been done manually (expand contrast/saturation, adjust "exposure" whites/blacks in color correction) or via using some normalization LUT.

Now, there are things like tone-mapping to normalize log media. Tone-mapping will look a bit different than any normalization LUT that may have been in use ... just like ALL normalization LUTs for any particular log media will be a bit different from each other.

I will say that in general tonemapping is the 'safer' way to normalize log encoded media. It doesn't get hammered by the clip being a bit more or less exposed than the LUT was built for handling. Which has always been a bugaboo with LUT based normalization practices.

Changing to working with tonemapped clips means of course doing some changes to the color corrections you will end up going with. Just as you have to do with  LUT based normalization. Just ... different.

But probably safer to your pixels.

Neil