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Using logarithms in the real world

C. Sivaram, Dec 4 2017, 23:47 IST
Many phenomena that occur in nature universally follow logarithmic laws. Representative image.

Many phenomena that occur in nature universally follow logarithmic laws. Representative image.

The English mathematician, Henry Briggs, in 1617, was responsible for the introduction of the so-called common logarithms (with the base 10). This was earlier called Briggsian logarithms. Following the pioneering publication of tables of logarithms by mathematician John Napier, Briggs consulted with him to propose an alternative definition of logarithms using a base of 10. In 1617, shortly after Napier’s death, Briggs published a logarithm table of the first 1000 numbers.

The beginnings

When Napier introduced logarithms, it simplified the task of many astronomers and others who spent much of their time doing tedious numerical computations. His mathematical interests, which were pursued in his spare time from church and state affairs, were in spherical trigonometry and in computation. It is not clear how he stumbled on the idea, but he probably took the hint from the well-known result that the product of two trigonometric expressions (such as two sines) can be found by finding the sum and difference of other expressions (involving cosines). As it is easier to add and subtract than to multiply and divide, these formulae produced a system of relation.

Again, he perhaps noticed that there exists a simple relationship between successive terms of a geometric progression and the corresponding exponents of the common ratio. He, at first, called the exponent of each power its ‘artificial number’ but later decided on the term logarithm, which means ratio number. The modern definition was introduced by Leonhard Euler, that is, if a number N equals b to the power of L (where b is affixed positive number other than one), then L is the logarithm (to the base b) of N.

Although Napier is presumed to have discovered the base E (called Naperian base), this is not true. It turns out that he came close to discovering the number 1/E, that is, the inverse of E. The number E is again universal number like N. It occurs universally in the laws of growth and decay of all physical systems. The mean life (of growth and decay) is defined as the time taken for the system to grow or diminish by a factor of E.

The concept of a base clearly took shape only with the introduction of ‘common’ (base 10) logarithms by Briggs in 1617. When Briggs met Napier, he proposed two modifications to make Napier’s tables more convenient, that is, to have the logarithm of one equal to zero and after considering many possibilities, Log 10 = 1 = 10 to the power of zero. In modern phrasing, this implies that if a positive number N is written N equals 10 to the power of L, then L is the common logarithm of N, written simply as log N. Thus, the logarithm of one billion is 9, one trillion is 12.

The words characteristic and mantissa were also suggested by Briggs. Thus, the logarithm of 200 is 2.3010, 2 being the characteristic and 0.3010 the mantissa. The logarithm of a product of several numbers A, B, C, etc is just the sum of logs of A, B, C, etc. Again, the logarithm of A raised to the power of N is just N log A. This simplifies tedious calculations. High powers or exponents can simply be converted to logs. Thus, Y equals log X to base A, if and only if X equals A to the power of Y. Thus, log X is the index to which a must be raised in order to get X. Logs to different bases can be related (logs to base E are called natural logarithms). Taking logarithms gives a sum of terms from a product of terms.

As Pierre-Simon Laplace, a scholar who worked in the fields of mathematics, statistics, physics and astronomy, remarked, “By shortening the labour, the invention of logarithms doubled the life of astronomers”. The invention of logarithms as far as dealing with complex calculation is concerned is second only in importance to the invention of the decimal and place value system (including the zero concept) pioneered by Indian and Arab mathematicians. As Lord Moulton said, “The invention of logarithms came to the world as a bolt from the blue. No previous work led to it or heralded its arrival. It stands isolated breaking in on human thought abruptly.”

Logarithms around us

It is remarkable that many phenomena in nature universally follow logarithmic laws. Measurement of sound intensity in decibels and size of earthquakes using Richter scale are some examples. Generally, this is quantified by the so called Weber-Fechner law, which states that the response (of the perceiving system) is proportional to the logarithm of the intensity of the stimulus. It is generalised to include any kind of physiological sensation, like perception of brightness caused by a source of light, or perception of loudness from a source of sound.

Our eye can see a candle 10 km away. The intensity of moonlight is a billion times higher, but the eye perceives it as only nine times higher (log billion is 9), so that moonlight does not scorch our eye. A 90-decibel loud conversation is a billion times greater than the very sensitive noise threshold of the ear, but the eardrum does not burst as the ear perceives it as only nine times more intense. The human ear is extremely sensitive to notice a change in pitch caused by a frequency change of only 0.2%. As a result, the musical notes that are written follow logarithmic scale on which the vertical distance (pitch) is proportional to the logarithm of the frequency.

Power laws are also ubiquitous in nature. The spectrum of high energy cosmic rays, radiation given off by charged particles at high velocities in a magnetic field, all follow power laws. Sights, sounds, smells, quakes, tsunamis, high energy radiation, growths, decays and many phenomena follow a logarithmic law. Entropy is a very fundamental concept underlying thermodynamics. It is defined by the Boltzmann formula, where entropy is the logarithm of total number of microstates of the system.

Logarithms are also involved in rocket dynamics in various forms. For a multistage rocket, the velocity and distance reached involve the logarithm of the mass ratio, that is, the ratio of the initial and final masses carried. The trajectory of a falling rocket is again a logarithmic spiral. In pure mathematics, again, logarithms are ubiquitous. The number of primes present below a given large number N is proportional to N divided by log N. Numerous examples can be given, underlying the universality of logarithms. Additionally, all our measurement scales are also based on logarithms.

With so many uses and applications in the natural world, it is hard to not notice that logarithmic laws universally govern nature.

(The author is with Indian Institute of Astrophysics, Bengaluru)

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