How do I find a natural log without a calculator?

For example, to approximate `ln(7)`, split the interval `[1, 7]` into a number of strips of equal width, and sum the areas of the trapezoids with vertices:

`(x_n, 0)`, `(x_n, 1/x_n)`, `(x_(n+1), 0)`, `(x_(n+1), 1/(x_(n+1)))`

If we use strips of width `1`, then we get six trapezoids with average heights: `3/4`, `5/12`, `7/24`, `9/40`, `11/60`, `13/84`.

If you add these, you find:

`3/4+5/12+7/24+9/40+11/60+13/84 ~~ 2.02`

If we use strips of width `1/2`, then we get twelve trapezoids with average heights: `5/6`, `7/12`, `9/20`,...,`25/156`,`27/182`

Then the total area is:

`1/2 xx (5/6+7/12+9/20+...+25/156+27/182) ~~ 1.97`

Actually `ln(7) ~~ 1.94591`, so these are not particularly accurate approximations.

Simpson's rule approximates the area under a curve using a quadratic approximation. For a given `h > 0`, the area under the curve of `f(x)` between `x_0` and `x_0 + 2h` is given by:

`h/3(f(x_0) + 4f(x_0+h) + f(x_0+2h))`

Try improving the approximation using Simpson's rule, using `h = 1/2` then we get six approximate areas to sum:

`h/3(25/6+73/30+145/84+241/180+361/330+505/546)~~1.947`

That's somewhat better.