Necessity may be the mother of invention, but it also takes that eureka! moment. By the way, the original eureka moment came from the ancient Greek mathematical genius Archimedes. He was struggling with finding a way to measure the volume of solids. This is easy for nice geometric shapes like cubes and spheres, but very difficult for irregular shapes like rocks. There is no formula for the volume of a rock! Legend has it that Archimedes figured it out while taking a bath. He noticed that the water level rose when he entered the tub, and then realized that he could find the volume of a rock by submerging it and seeing how much water was displaced. He shouted eureka! and ran naked through the streets of Syracuse in joy.
The necessity that drove the discover of logarithms was due to the fact that multiplying and dividing large numbers by hand is extremely difficult. It was so difficult that it held back Kepler’s work in astronomy for years. Here is the eureka part of the discovery of logarithms.
Look at the following infinite series:
1, x, x2, x3, x4, x5, x6, …
In order to multiply any two terms in the series, you simply add the exponents. This turns multiplication into addition, and division into subtraction. That is a huge increase in the speed of calculation. There is only one catch, which is that you have to find a way to represent numbers using the same base.
But even using a number as small as 2, this is not possible. What if you have to multiply 3 by 100? You need to do one of two things:
Method One: Express numbers as fractional exponents. Thus
3 × 100 = 21.5849 + 26.6438
= 28.2287
= 300
This is the approach we use now, although we use the natural logarithm, e, as our base rather than 2. But people did not know how to use fractional exponents back when logarithms were first developed. So they used the second approach.
Method Two: Choose a really, really small base. One of the first bases chosen was 1.0001. Using this approach we get
3 × 100 = 1.000110986 + 1.000146054
= 1.000157040
= 300
I left off one really important step, which is finding the logarithms of 3 and 100 to the base of 2 or 1.0001. But there is no magic here. In olden days you simply looked these values up on a chart. Of course, working out the values on those charts was a time consuming process that took years. Log tables were worth more than their weight in gold. Eventually slide rules replaced charts. Nowadays we simply use a calculator.
Exercises
- multiply 3 by 100 using natural logs.
