It turns out that the common logarithms of each of the numbers from 2 through 9 can be very well approximated by rational numbers of the form n/40.

Derive each of the numerators with no calculation aids beyond pencil and paper.

(In reply to Who is right for 7? by Jer)

 

Jer,

34 is better.

I did not post  my solution, because I was intrigued by another problem, triggered by this one.

"Calculate the logarithms, not  necessarily  with the denominator equaling 40, but warranting better precision."

 

The equations  I've used give better approximation , even starting with  log2=.3

 

LOG2  defines values of LOG4 , LOG5   & LOG8.

For x=LOG7 Iet's  start with  9886633715=7^11*5== very close to 10^10

 

 so 10=11x+.7 and x=9.3/11= .845454545

    btw  40*x=33.818, i.e. closer to34

MY LOG3 was  obtained from 

 2401=7^4= very close to 100*8*3.

  4*.845454545 =2+.9+LOG3  and  LOG3=.481818 

btw  40*LOG3=19.2727   .

I will not bore you with more details, but I have compared all the values with the  correct numbers, evaluated their  accuracy and calculated the interpolation values.

So , if   I ever  will be posted  incommunicado on some isolated island with nothing , but p&p and an urgent need to calculate log 1234.5- I WILL MANAGE!!

That was a lousy example - (I would  interpolate between 1200 and 1250- trust me....)

Edited on June 1, 2011, 7:31 pm

Posted by Ady TZIDON on 2011-06-01 19:23:47