The logarithmic utility function has no boundary value. The curve in the graph appears to flatten toward the upper right. But really, it never stops growing. This means, for example, that a person with logarithmic utility would be equally happy to receive any amount of money that would increase his or her wealth tenfold. Going from $10,000 to $100,000 or from $100,000 to $1 million is the same as going from $1 million to $10 million.
This may be reasonable or unreasonable. There is one thing to consider here. Does owning $10 billion have any advantages over owning $1 billion? The answer is no if you only care about "living well". So is owning 10 billion billion dollars more proud than owning 1 billion billion dollars? The answer is still no if you only care about becoming the richest person in the world.
The application of logarithms is also not a good model for representing poverty because it implies that losing 90% of your last million is as unfortunate as losing 90% of a cent. Finally in your pocket. This sounds really odd.
