I get that your utility function from money is non-linear, but I would expect a more accurate model to be a step-function, with large steps at "out of debt", "can tell a bad boss sayonara", "can buy a house", "can pay for kids' college eduaction", and "never have to work again". Equity payouts from a typical startup exit often line up nicely with the middle three, and if you hit the Google/Facebook jackpot, sometimes the last.
As others have mentioned, maximizing log is equivalent to maximizing the underlying.
But when considering returns on accruing capital, a 20% loss is much worse than a 20% gain is good, and similarly a 100% gain is much less good than a 100% loss is bad. This is correctly captured by taking the log. In the case where money is simply accrued from some external source, and their is no proportional return, log isn't necessary.
Independent investment opportunities generally have the effect of multiplying your value by a random amount over a specified interval. When you take logs, you are adding a random amount instead. From the strong law of large numbers, after enough intervals, it is statistically certain that the sum of the logs of those random numbers numbers will converge on the number of intervals times the expected value of the log of your investment strategy.
Therefore maximizing the expected value of the log maximizes the long term rate of return that you (with 100% probability) will observe.
I think you just answered your own questions - log is continuous.