# Question on Logarithms

1. Jun 25, 2011

### assuredlonewo

How were logarithms calculated before the use of calculators.

2. Jun 25, 2011

### gb7nash

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.

3. Jun 26, 2011

### SteveL27

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

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

4. Jun 26, 2011

### 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

5. Jun 26, 2011

### Integral

Staff Emeritus
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.

6. Jun 26, 2011

### HallsofIvy

Oh, dear! You are showing your age!

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

(Before Calculators)

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