Numbers and algebra
1614
Logarithms
John Napier; tables in base ten by Henry Briggs
What if multiplying were as easy as adding?
Astronomers spent weeks multiplying huge numbers by hand. Napier found a way to turn multiplication into addition, using tables of logarithms. Kepler soon built his planetary tables on them. Laplace later said logarithms doubled the life of the astronomer.
First, make a guess
A slide rule multiplies two numbers by…
- a (under the blue 1)
- 2.000
- b (on the blue scale)
- 3.000
- a × b (on the fixed scale)
- 6.000 (exact: 6.000)
Each scale is marked by logarithms, so distances add when numbers multiply. Slide the blue scale so its 1 sits over a number a on the fixed scale, then move the pink cursor to b on the blue scale: the fixed scale shows a × b. Engineers multiplied this way for over 300 years, including for the Moon landings.
How they figured it out
John Napier, a Scottish landowner, spent about twenty years computing his tables by hand. He imagined two points moving: one along an endless line at steady speed, the other along a fixed segment, slowing down in proportion to the distance it had left. Matching the two motions gave him logarithms.
In 1615 the mathematician Henry Briggs travelled from London to Edinburgh to meet him, and they agreed base 10 would be easier. Briggs produced those tables. Within a few years Kepler was using logarithms for his planetary tables, and navigators, engineers and astronomers used them for three centuries.
The equation
To multiply two numbers, add their logarithms, then look the answer back up.
See it in real life
Scales that squeeze huge ranges
A whisper is about 30 dB and a rock concert about 110 dB: 80 dB more means 100 million times the sound intensity. A magnitude 7 earthquake releases about 32 times the energy of a 6. Every piano octave doubles the frequency, so a keyboard is a logarithmic scale you can play.
Where engineers use it
Decibels
Sound levels and signal gains are logarithmic: +10 dB is ten times the power.
Log plots
Bode plots, Richter scales and pH all squeeze huge ranges onto one readable axis.
Worked example: Multiplying by adding
- and
- Add them:
- , so .
What it led to
- Euler’s formula 1748
Led to
Euler identified e as the natural base of logarithms and joined it to the circle.
- The age of the Earth 1956
Built on
Reading an age from a ratio means undoing an exponential: a logarithm.