Common misconception. It does not.

It makes only two assumptions:

- Growth compounds, and

- More is better.

Under those two assumptions, log-dollars is what you need to optimise.

By coincidence, logarithmic utility of money would also lead to the same conclusion, but that's mathematical happenstance, and not something going into the model.

----

Another way to put it: the Kelly criterion prescribes logarithmic utility. It says that to maximise growth (under above assumptions) you ought to adopt logarithmic utility. If you don't, you get worse results.

Looking at the wikipedia derivation of the Kelly criterion, taking the logarithm of the expected rate of return E[r] = (1 + fb)^p (1 - fa)^(1-p) is just to make finding the derivative easier (because logs are monotonic). Nothing else.