The transfer functions in your (rec.709) color space is non-linear indeed. However, the pixel values you store are in a linear relationship with each other. The difference between values 20 and 21 are the same as the difference between values 120 and 121, assuming an 8bit signal. I.e., the information is the same for all pixels. Further down the chain, these values are then mapped onto a gamma curve, which is non-linear.

What the "log"-spaces are doing is to use a non-linear relationship for the pixel values, as a form of lossy compression. If the signal has to factor through 8bit values, using a compression scheme before it hits the (final) gamma curve is a smart move. If we retain less precision around the low and high pixel values and more precision in the middle, we can get more information from the camera sensor in a certain region. Furthermore, we can map a higher dynamic range. It often looks more pleasing to the eye, because we can tune the setup such that it delivers a lot of precision and detail where our perception works the best.

In short: we are storing (8bit/10bit) pixel values. The interpretation of these values are done in the context of a given color space. In classic (rec.709) color spaces, the storage is linear and then mapped onto a non-linear transfer function. In the "log" spaces, the storage is non-linear and is then mapped onto a non-linear transfer function. In essence we perform lossy compression when we store the pixel in the camera.