The mystery behind LOGS!!!?

Question

Just out of curiosity, is there any way to figure out logs of numbers without the use of a calculator, say log[base10](758585) or something simpler? I mean, before calculators, how did mathematicians solve logarithms (other than just having lists - and how did they make these lists)? Also, how do calculators do it (this is not supposed to be a stupid question); I'm guessing there is some sort of matrix or something; can't we apply that on our own -- only it would take us MUCH, MUCH longer?

Thanks,
AC

And I'm not saying estimated logs; I mean, EXACT answers to logs; how do you do it?

Answer 1

I think they use the Logs Rules.


the easiest way is to use the basic rule of Log which you know is
when you say Log A(base on C)= B it means that " C^B=A"(C power B equals A)
for example you know that 10^2=100 so Log 100(base on 10)=2
i think they do it in this way !!!

Answer 2

logs that are not exact are often irrational and so cannot be determined exactly to the last digit.

One way to calculate logs is from the Taylor Series for the f(x)=log(x) function. http://mathworld.wolfram.com/TaylorSeries.html

Another way is by linear interpolation (educated guessing, and then narrowing in on successive guesses).

Another way is mechanically, by constructing a reverse slide rule based on a few known values, and using a slide to perfrom the interpolation.

Answer 3

They did it using infinit series. For example, for natural logs,
ln(1+x) = x-x^2/2+x^3/3 -x^4/4 + ...+(-1)^(n+1)*x^n/n + ....

Usually these series converge quickly meaning that after 5 or six terms you have an accuracy of 5 places.

There are series for pi, e, sinx, cosx, etc.