The joy of receiving X amount of money is proportional to ln X

This makes sense. When you have a little money, some more money is joyful. When you have some money, some more is nice. When you lots of money, some more is not that important. When you are extremely rich, you couldn't care less for just "some" more. The use of the natural logarithm sets the pace at which you become uniterested in money.

Mathematically we can study how happy I would be for twice as much money:

Joy_of_Money(2X) = ln 2X = ln 2 + ln X

As you see, for small X the contribution ( ln 2 ) to my happiness is big, while as for large X, it is tiny.

Or expressed graphically:

```

joy of money

A
|
|                                                .      .
|                                 .
|                       .
|                .
|           .
|        .
|      .
|     .
|    .
|    .
|   .
|   .
|  .
|  .
|  .
| .
| .
| .
|.
|.
|.
|.
+-----------------------------------------------------> money
```
It is hard to draw the natural logarithm in ASCII, but you get the idea. The real f(x)=ln(x) curve never really becomes horizontal.

This idea was to my knowldege first expressed by a professor of mathematics at my university. Jennifer pointed out to me that this is related to law of diminishing returns, which it is.