Clarify logarithm discussion

[keiravillekode]

[no important files changed]

[keiravillekode]

https://www.efunda.com/math/taylor_series/logarithmic.cfm shows where various series converge.

When people say the Taylor series for log of 1 + z converges for z between +/- 1, that is equivalent to saying the Taylor series for log of x converges for x between 0 and 2. To see this, set x = 1 + z

[keiravillekode]

In our instructions, log always refers to the natural (base e) logarithm. There is also a common convention where ln means the natural logarithm and log means the base 10 logarithm, but our instructions are referring to natural logarithms throughout and therefore are not using that convention.

Using log for natural logarithm is certainly common:

• JavaScript
• C and C++
• Julia has both log and log10

If we were interested in base 10 logarithms, the formula at the very top of our instructions could be written as
x ^ y = 10 ^ (y * log10(x))

But this is not what we are interested in.

Another option would be for our instructions to show ln instead of log

[keiravillekode]

I missed the original point in the part where you switched to using ln

Now updated with a different way to compute logarithms for larger x

[keiravillekode]

Changed to use ln

The part I removed gave 10 special status, when that is not necessary. For example, why are we not choosing 2 or exp(1), i.e. e.

If we have a value larger than 1, we can always take the reciprocal to give a value less than 1. I think this is more elegant.

#! /usr/bin/env python3

import math

numbers = [0.001, 0.002, 0.005, 0.01, 0.02, 0.05, 0.1, 0.2, 0.5, 1, 2, 5, 10, 20, 50, 100, 200, 500, 1000]

def bilinear(z):
    n = 1
    term = (z - 1) / (z + 1)
    ratio = term * term
    previous = 0
    total = term
    while previous != total:
        previous = total
        term = ratio * term
        n += 2
        total += term / n
    return 2 * total

def taylor(x):
    if x > 1:
        return - taylor(1 / x)
    z = x - 1
    n = 1
    power = z
    previous = 0
    total = power
    while previous != total:
        previous = total
        power *= -z
        n += 1
        total += power / n
    return total

for number in numbers:
    actual = math.log(number)
    b = bilinear(number)
    t = taylor(number)
    print(f"{number}: actual {actual}, bilinear {b}, taylor {t}")