I'm sorry, but you are plain wrong. The log has nothing to do with utility. And there is no chance of really understanding the result if you're confusing yourself with that bad idea.
To start, EVERY utility function that is both increasing and sublinear will agree that Kelly is the best strategy. Whether square root, log, or bounded - it doesn't matter. The details of your utility function are unimportant.
What matters is that each iteration of an investment strategy multiplies your net worth by a random factor. But log turns multiplication into addition. And statistics has very strong results about sums of independent variables.
The result is that with 100% odds, a player following Kelly will eventually wind up ahead of any other static strategy that you could choose. Both wind up ahead and eventually remain ahead. Which is why a wide variety of utility functions will conclude that Kelly is the optimal strategy.
To start, EVERY utility function that is both increasing and sublinear will agree that Kelly is the best strategy. Whether square root, log, or bounded - it doesn't matter. The details of your utility function are unimportant.
What matters is that each iteration of an investment strategy multiplies your net worth by a random factor. But log turns multiplication into addition. And statistics has very strong results about sums of independent variables.
The result is that with 100% odds, a player following Kelly will eventually wind up ahead of any other static strategy that you could choose. Both wind up ahead and eventually remain ahead. Which is why a wide variety of utility functions will conclude that Kelly is the optimal strategy.