Natural logarithm of 2 [HP-15C/HP-42S/Free42 & others]

[C.Ret]

(10-12-2019 01:30 PM)ttw Wrote:  Perhaps the methods used here may be of interest.

http://elib.mi.sanu.ac.rs/files/journals...tm1212.pdf

In this publication, the five point Gauss-Legendre Log (5P-GLLOG) lead to this approximation :

\( ln(1+x)\approx \frac{7560x+15120x^2+9870x^3+2310x^4+137x^5}{7560+18900x+16800x^2+6300x^3+900x^4​+30x^5} \)

One may deduce that
\( ln(1+1)\approx \frac{7560+15120+9870+2310+137}{7560+18900+16800+6300+900+30} \)
\( ln(2)\approx \frac{34977}{50490}=0.\underline{6931471}57853 \)

A better fraction may be proposed only base of the numeric value without any further mathematical or fundamental consideration:
\( ln(2)\approx \frac{261740}{377611}= 0.\underline{6931471805}641255154113624867919631578529227167640773176628858799 \cdots \)
Best numeric approximation I get on my HP-71B using FRAC$(LOG(2))
which coincidentally correspond to
\(ln(2)\approx[0;1,2,3,1,6,3,1,1,2,1,1,1,1,3,10]\)