Calculating Logarithms Before Calculators: History & Methods

[assuredlonewo]

How were logarithms calculated before the use of calculators.

 

[gb7nash]

Here's an iterative method to approximate it. For instance:

Assume you want to calculate log_ab, where a and b are positive numbers. So you want to find x such that a^x = b. From visual inspection, pick a point y so that you're sure that a^y < b and a point z so that a^z > b.

Now consider w = (y+z)/2 (which is the midpoint between y and z). One of three things will happen:

1) a^w < b

2) a^w > b

3) a^w = b

If a^w < b, take the midpoint between w and z and repeat. If a^w > b, take the midpoint between w and y and repeat. If a^w = b, then you're done(though this will probably not happen).

Keep repeating until you're within a certain epsilon of b.

_____

You can also look at the taylor series expansion around a certain point and cut it off past a certain point. This might be more work though.

 

[SteveL27]

How were logarithms calculated before the use of calculators.

Before electronic pocket calculators became common, every engineering student owned one of these ...

http://en.wikipedia.org/wiki/Slide_rule

 

[HallsofIvy]

Yes, but you needed to know the value of the logarithms in order to make a slide rule.

I can't speak for what was done historically, but you could use the Taylor's series for the logarithm:
$$ln(x)= \sum_{n=1}^\infty \frac{(-1)^{n-1}}{n}(x- 1)^n$$

Ahh!
On "Math Forum- Ask Dr. Math"
http://mathforum.org/library/drmath/view/52469.html
they have

Instead of taking powers of a number close to 1, as had
Napier, Briggs began with log(10) = 1 and then found other logarithms
by taking successive roots. By finding sqrt(10) = 3.162277 for
example, Briggs had log(3.162277) = 0.500000, and from 10^(3/4) =
sqrt(31.62277) = 5.623413 he had log(5.623413) = 0.7500000.
Continuing in this manner, he computed other common logarithms.
Briggs published his tables of logarithms of numbers from 1 to 1000,
each carried out to 14 places of decimals, in 1617.

 

[Integral]

If you needed more then the 3 digit accuracy of a slide rule you opened a book of log tables. We were even taught to do a linear interpolation to get values between tabulated values.

Halls post is the answer to how did they generate the tables.

 

[HallsofIvy]

Oh, dear! You are showing your age!

Yes that's how we did it in the years "B.C.".


(Before Calculators)