I think the log is just the utility function. You could substitute it with any non-linear utility function and it would've gave you another answer that makes sense for that utility.
The main takeaway is that a linear expected utility doesn't make sense.
It would've told you to bet all your wealth every game, which does result in a higher linear expected value, where you win (1+1.1)^N with probability 1/2^N at time N, but 0 otherwise. But no real human would take the bet of extreme high payoff at extremely rare chances with ruin otherwise.
Also see St. Petersburg paradox for a similar "paradox" resolved with expected utility theory.
The main takeaway is that a linear expected utility doesn't make sense.
It would've told you to bet all your wealth every game, which does result in a higher linear expected value, where you win (1+1.1)^N with probability 1/2^N at time N, but 0 otherwise. But no real human would take the bet of extreme high payoff at extremely rare chances with ruin otherwise.
Also see St. Petersburg paradox for a similar "paradox" resolved with expected utility theory.