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I have no idea what function it is. I also don't really have a working understanding of what "logarithmic" is. It starts falling because when you're dealing in the thousands of dollars, the next dollar matters less than it did when you were dealing in the tens of dollars.
Oh, okay, I think we're talking about the same function in different terms. You're talking in terms of the utility function itself, and I was talking about how much the growth rate falls as the amount of money decreases from some positive starting point, since that's what Hul-Gil seemed to be talking about. (I think that would be hyperbolic rather than exponential, though.)
The utility function itself does grow indefinitely; just really slowly at some point. And at no point is its own growth speeding up rather than slowing down.