Question

How do you calculate `ln(1.76)`?

Answer 1

`ln 1.76 = int_1^1.76 1/t dt ~~ 0.565`

Explanation:

Hmmm. I'm not sure exactly what we are looking for here.

We could try using the series:

`ln(1+t) = t-t^2/2+t^3/3-t^4/4+...`

but it would converge somewhat slowly.

We could evaluate the integral:

`int_1^1.76 1/t dt`

using one of the popular approximation methods.

For example, Simpson's rule tells us that:

`int_a^b f(t) dt ~~ (b-a)/6 (f(a)+4f((a+b)/2)+f(b))`

So putting `f(t) = 1/t`, `a=1` and `b=1.76`, we find:

`ln 1.76 = int_1^1.76 1/t dt`

`color(white)(ln 1.76) ~~ (1.76-1)/6(1/1+4/((1+1.76)/2)+1/1.76)`

`color(white)(ln 1.76) ~~ 0.76/6(1+8/2.76+1/1.76)`

`color(white)(ln 1.76) ~~ 0.565`