Counting Digits

The ubiquitous logarithm

Like astronomy and atomic physics, information technology must grapple with numbers too large for human comprehension. They can be brought down to a manageable level by the use of a mathematical curve called the natural logarithm. Since its invention in the seventeenth century, when it vastly increased the computational power of astronomers, the logarithm has become an indispensable practical tool in science, cropping up in a wide variety of different fields and hinting at formal connections where none would be suspected. Inasmuch as many of the things the logarithm describes impinge on our daily lives, we would do well to learn its lessons and to incorporate them into our thinking.

On graph paper, the logarithm and its inverse, the exponential, a smooth curve through the points of the geometric series, mirror each other across the diagonal line from the lower left to the upper right. Viewed in one direction, the exponential doubles, and then doubles again towards infinity; in the other, it shrinks and halves, approaching zero but never reaching it. The exponential’s breakneck rise describes some of the characteristic phenomena of our time: the information explosion, the hydra-head proliferation of AIDS viruses, the development of embryo by cell division, the growth of money by compound interest, and the runaway chain reaction of fissionable material in an atomic bomb. At the same time, its diminishing tail characterizes both radioactive decay and the fading peal of a church bell.

Like the exponential, the logarithm always rises from left to right. But whereas the exponential roars unchecked to infinity at an ever-increasing rate or slope, the rise of the logarithmic function is accompanied by a slope that gets continuously flatter; and whereas the exponential approaches the horizontal axis to the left of the origin like a landing glider that never quite touches down, the logarithm plunges precipitously through zero to negative infinity with increasing steepness, hugging the vertical axis ever more closely, though never quite reaching it.

At their extremities the exponential and logarithmic curves diverge dramatically, but near the origin they approach each other, running momentarily in parallel. Together they form the graceful outline of the thin waist of an hourglass.

The utility of the logarithm is aptly demonstrated by its ability to represent numerical excesses in comprehensible terms. To see how that works, consider a close cousin of the natural logarithm, called the logarithm to the base ten, or in its abbreviated form, the log. For multiple of ten, the log simply counts zeros, recording them as positive when they appear in the numerator and negative when they appear in the denominator. Thus the log of 1000 is 3 (written 1000 = 103), the log of 1/100 is −2 (written 10 = 10−2), and the log of 1 is zero (written 1 = 100). More generally, the log counts the digits of a given number, at least approximately. The rule is easy to remember: if you come across a number larger than one, written in ordinary decimal notation, you can find the approximate value of its log by simply counting its digits to the left of the decimal point.

A plot of the log has remarkable properties. On graph paper divided into one-centimetre squares, a point on the curve a mere eleven centimetres (or half a page) above the horizontal axis, lies 100 billion (1011) centimetres to the right — reaching past the orbit of the Moon. Conversely, at a point eight centimetres below the origin the curve has moved to within 100 millionth (10−8) of a centimetre, or an atom’s diameter, from the vertical axis.

Scientists long ago adopted those properties with the ‘powers of ten’ notation. It is a marvelously convenient shorthand — one that prevents errors and saves space. Imagine calculating the properties of the early cosmos in longhand, writing out in full decimal notation the values of the Planck time (10−43 seconds) and the Planck length (about 10−35 metres) with their combined total of seventy-eight zeros — an impossible feat of patience and care.

Beyond this, the log further simplifies computations by converting arithmetic into counting digits. For example, 600 times 6000 equals 3,600,000 but three digits plus four digits equals seven digits. So if you are only interested in rough approximations, and a count of its digits is a good enough indicator of the magnitude of a number, then you never have to learn to multiply. Division is similarly simplified by becoming subtraction. This marvelous property of the log is responsible for the design of that ancient trademark of the engineer, the slide rule, which achieves multiplication and division of number by mechanically adding and subtracting their logs. Although the slide rule has now become obsolete, the log is still very much with us as an effective analytical tool for understanding the world, a way of looking at things that might be called ‘logarithmic thinking’, or more simply ‘power thinking’.

Information: The New Language of Science

Hans Christian von Baeyer