Calculate logarithm witout calculator

[Guest]

Hi! I have to calculate log 9 to base 2 without using calculator. There is an algorithm ,described on wikipedia,but I can not apply it ....could you guys help me? Thanks!

http://en.wikipedia.org/wiki/Binary_logarithm Recursive approximation.

[Guest]

I will just use log to mean log base 2.

log 9 = log (8*(9/8)) (note that 8 is a power of 2 and 9/8 is between 1 and 2)
= log 8 + log (9/8)
= 3 + log (9/8)
The following steps we will repeat:
= 3 + log ((9/8)^8) / 8 (note that (9/8)^8 is between 2 and 4, and 8 is a power of 2)
= 3 + log (43046721/16777216) / 8
= 3 + log (2*43046721/33554432) / 8 (note that 43046721/33554432 is between 1 and 2)
= 3 + (log 2 + log 43046721/33554432)/8
= 3 + (1 + log 43046721/33554432)/8
= 3 + 1/8 + (log 43046721/33554432)/8
Just like 9/8 was between 1 and 2 so is 43046721/33554432. We can repeat the procedure on it to get a better approximation.
= 3+1/8 + ( log ((43046721/33554432)^4) / 4 )/8 (note that 4 is a power of 2 and (43046721/33554432)^4 is between 2 and 4)
= 3+1/8 + log ((43046721/33554432)^4) / 32
= 3+1/8 + (log 2 + log ((43046721/33554432)^4 / 2))/32
= 3+1/8+1/32 + log ((43046721/33554432)^4 / 2))/32
Just like 9/8 and 43046721/33554432 were between 1 and 2 so is (43046721/33554432)^4 / 2, so we can repeat the procedure to get a better approximation (but I won't as the numbers are getting large now, it might be better if you are doing things by hand to convert the fractions to decimals and just drop the digits after a few decimal places to make things more manageable, for example 43046721/33554432 = 1.282892256975174 = 1.2829 approximately).

So from our calculation log 9 is approximately 3+1/8+1/32 = 3.15625, in fact we can say it lies between 3.15625 and 3.15625+1/32 = 3.18750 (using a calculator the actual answer is 3.169925...)

I've only done 2 iterations but if you do a few more (using decimals instead of fractions) you'll get a much better answer.

Hope this helped,

R. Baber.