Using ln as our utility, at time N we get: E[ln(X_1*...*X_N)] = sum E[ln(X)] So kelly happens to maximize this expected utility by construction since it was

Using ln as our utility, at time N we get: E[ln(X_1*...*X_N)] = sum E[ln(X)] So kelly happens to maximize this expected utility by construction since it was derived by maximizing expected log of one round of betting (E[ln(X)]). But if we use square root as our utility instead: E[sqrt(X_1*...*X_N)] = prod E[sqrt(X)] We would maximize this expected utility by maximizing E[sqrt(X)]. Going through the same calculus we can see that we don't arrive at kelly. Where did I go wrong?