The logarithmic utility function makes the Kelly criterion optimal even when there is only one remaining bet. For example say you have $x, the probability of winning is p, and you bet $y. Then you want to maximise

    plog(x+y) + (1-p)log(x-y).

Setting the derivative with respect to y equal to 0 yields

    p/(x+y) - (1-p)/(x-y) = 0.

This rearranges to give

    y = (2p - 1)x,

which is precisely the Kelly criterion.