This bit is miscellany about Approximating logarithms theres also a link at the foot of the page to some elementary descriptions of how to make a continued fraction which might be helpful given that i used a variant of this method to make an approximation program

Numbers in art is intended to be a collection of ideas on their use based on some ancient programs written by Mike Burr.

Approximate logarithms

I'm not going to describe the algorithm in detail as its doesn't lend itself to that
but suffice it to say you keep reducing the power term while extracting a nearest residue
in roughly the same way as per the continued fractions
- and i hope to load this program onto the web as its extremely versatile
and quite intriguing.
Heres a sequence using the number base of 10 and is an approximation to log 13
from above
13 :- 1/0 ; 2/1 ; 3/2 ; 4/3 ; 5/4 ; 6/5 ; 7/6 ; 8/7 ; 9/8 ; 19/17 ; 29/26 ; 39/35 ;
127/114 ; 215/193 ; 303/272 ; 391/351 ; 479/430 ; 567/509 ; 655/588 ; 1398/1255 ;

approximation using last ratio = 1.11394422310756972112-1.11394335230683676921
resulting in an error of about 8.7e-7 - incidentally the approximation from below is quicker
giving an error about 3.3e-17
In a similar way i used the (bottom line of the ratios) to construct a test picture
made with kids coloured pencils and which i still have somewhere here.
Heres a table of approximate logarithms .. for a load of sequential numbers as a foot note here are their continued fraction representations log table continued fractions .. note that sometimes the number is reduced to zero and this i annotated in the output [ which looks odd at first ]


Ruby program "calllogit.rb" used to generate single dimension logarithmic approximation