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?
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