The traditional resolution of the St. Petersburg Paradox involves adding a utility function to the problem, taking into consideration diminishing marginal utility.

If the utility of winning $\(n\) is a logarithmic function, specifically \(\log(n)\), then what is the expected payout, factoring in utility, that a player would get from playing the St. Petersburg Paradox game? Put another way, if the value of \(n\) diminishes \(\log(n)\), what is the break even point such that a player should only pay less than this answer to play the game?

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