Logs of Nature

27 November 2012

We generally think of numbers as a linear progression from 1 to 2 to 3, etc. We also tend to measure things around us on a linear scale. A 10 hour road trip, for example, is very different from a 1 hour one. In the sciences, however, it is often more useful to look at things on a logarithmic scale.

A logarithm scale is one that focuses on the overall size, or “order of magnitude” of objects. For example, if something has a mass of 100 kilograms, then on a log scale it would be 2, since 100 = 10 x 10. Likewise, 1,000 would be 3 on a log scale, since 1,000 = 10 x 10 x 10. In general, you can base your log on any number you want. The number 128 is 2 x 2 x 2 x 2 x 2 x 2 x 2, so you can say its log is 7 in “base 2.” Perhaps the most common base in the physical sciences is the so-called natural log, which is a log of base e, where e is an irrational number about equal to 2.71818…

Log scales are so deeply rooted in physical phenomena that even our eyes and ears operate on a logarithmic scale. This is why the loudness of sound is measured in the logarithmic decibel scale, and the brightness of stars is measured in apparent magnitude, which is a logarithmic scale of luminosity. Even young children tend to perceive numbers on a logarithmic scale before we teach them linear counting.

Perhaps the most famous demonstration of a logarithmic scale is the short movie Powers of Ten, by Charles and Ray Eames, seen above. While it’s a bit outdated, it shows how the universe can be viewed on a logarithmic scale, and how on such a scale humans exist roughly in the middle of the very large and the very small.