# Question on Logarithms

## Main Question or Discussion Point

How were logarithms calculated before the use of calculators.

gb7nash
Homework Helper
Here's an iterative method to approximate it. For instance:

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

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

1) aw < b

2) aw > b

3) aw = b

If aw < b, take the midpoint between w and z and repeat. If aw > b, take the midpoint between w and y and repeat. If aw = 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.

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
Homework Helper
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
Staff Emeritus
Gold Member
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