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  The Log utility function is a special case of the Power utility, corresponding to the case gamma = 0.
If wealth( s) is your wealth outcome under scenario s,  then
 utility( wealth(s)) = log( wealth( s)).
     The Log utility risk measure is the extreme case ( most risk preferring) of the Power utility risk measure.

 Kelly Criterion
    Having the goal of maximizing the long term growth rate is known as the Kelly Criterion.
Kelly showed that maximizing the log( end of period wealth) each period is consistent with this goal.
So using a log utility for risk is sometimes known as using the Kelly Criterion.
   Suppose there are 1000 other investors, who each period invest so as to maximize their expected wealth at the end of the period.
You, however, invest so as to maximize the expected log( end of period wealth).
After a few periods, the following will be true with high probability:
   a) the average wealth of the 1000 other investors will be greater than yours,
   b) you will be wealthier than almost all of the 1000 other investors.
Essentially, most of the 1000 other investors will be almost broke, and one or two of them will be very wealthy.
     The simplest use of the Kelly Criterion is in bet sizing, i.e., what fraction of our current wealth should be
put a risk in a bet (on the market) vs. what fraction should be kept in cash for future bets.
Kelly portfolios are superior to traditional strategies such as mean variance in the long run,
however,  in a single period they lack diversification and have high volatility.