Exponential growth is growth which occurs at a rate proportional to the size of the thing growing; exponential decay is likewise decay which occurs at a rate proportional to the size of the thing decaying. These two ideas are enormously important in science, and indeed in life in general; there are many things which it is just not possible to understand without grasping exponential relationships.

The classic example of exponential growth is the growth of bacteria; in ideal conditions, each bacterium in a population will divide into two, so the total population will double every generation (with a period of about twenty minutes in the case of some bacteria). The increase in the power of the worlds' computers is often cited as another example of exponential growth; the faster our computers, the more power we have to make them even smaller and faster, hence Moore's Law, which suggests that the power of the fastest chips doubles roughly every eighteen months. According to the inflationary universe theory of cosmology the universe underwent a brief, extremely rapid period of exponential expansion about 10-32 seconds into its existence, blowing up to about the size of a tennis ball. Exponential growth of real quantities will inevitably hit limits; bacteria will exhaust their food supply, chip manufacturers will hit up against problems caused by quantum noise, and so on. This is a problem which need not affect made-up quantities like money, however, so the exponential growth in the cost of a loaf of bread caused by steady inflation could in principle continue indefinitely.

The classic example of exponential decay is radioactive decay; each radioactive atom in a sample has a fixed chance of decaying in any given time period, so the rate of decay is always proportional to how much radioactive material remains. This is why we talk about the half-life of a radioactive isotope; this is the time that it takes for half of its unstable atoms to decay. After another half-life, half of the remaining atoms will then have decayed, which is to say another quarter of the atoms that were there in the first place; after three half-lifes one eighth will be left, and so on. The charge in a capacitor also decays exponentially, since the rate at which it discharges is proportional to its voltage, which in turn is proportional to the remaining charge. Other fun examples of exponential decrease include the amount of water left in an emptying bathtub, and the decay of the head on a pint of beer.

Another realm in which the idea of an exponential (or logarithmic) relationship is crucial is human perception; in many areas, our responses to stimuli are logarithmic, not linear in nature. This means, for instance, that we perceive a series of notes in which each note is some multiple of the frequency of the previous note as being evenly spaced - this is why each octave starts at twice the frequency of the one before, rather than being evenly spaced along the frequency spectrum. We also perceive the loudness of sound logarithmically, and the decibel scale reflects this; a difference of ten decibels represents a doubling in the intensity of a sound, but subjectively the difference between a 140 dB sound and a 130 dB is no greater than the difference between a 100 and a 110 dB sound. This is certainly true of our visual response, as well - an exponential increase in brightness is perceived as if it were a smooth increase - and it may well be true of all of the senses. It is not hard to see why this would be the case; from the point of view of survival, it is generally much more important to be able to respond to a wide range of stimuli strengths (including very low levels) than it is to be able to accurately perceive small differences in large stimuli.

In general, an exponential function is any function of the form f(x)=ax - exponent is another word for index or power in this context. The inverse of an exponential is a logarithm, which is to say that, for instance, log10 (10x) = x. This is usually written simply log (10x) = x since log is assumed to refer to the base-10 logarithm unless otherwise specified (although I am told that it sometimes means ln, which I'm just about to get to). To get a feeling for what this means, consider that for any power of ten - 10, 100, 1000, etc. - its logarithm is equal to how many zeros there are in the number. It is possible to take a logarithm in any base, but calculators generally only provide two kinds of logarithm - the log function mentioned already, and the natural logarithm (written ln), which is the logarithm with base e, an irrational number roughly equal to 2.718. The reasons why e and its logarithm are of interest to mathematicians are intriguing in themselves, but they needn't concern us here. More relevant right now is the point that it is pretty easy to convert between logarithms of different bases, say a and b:

logb x= loga x / loga b

All exponential growth and decay is governed by the equation

where N is the exponentially-changing quantity, N0 is its value at time t=0, and k is the growth constant or decay constant. It is quite possible to replace e here with another number - 2 or 10, for instance - by scaling the growth/decay constant accordingly, but using e makes the mathematics a little more elegant. Since the logarithm is the inverse of the exponential, it is possible to re-arrange this equation by dividing both sides by N0 and then taking the natural logarithm of both sides to get
ln (N/N0)=kt
This is useful because we can then do things like plugging N/N0=1/2 into the equation, in order to calculate the half-life of something from its decay constant (which, incidentally, is usually called λ in the context of radioactive decay).

Be aware that if a rate of change is exponential, that says nothing at all about how fast or slow it might be right now; some isotopes, for example, have half-lives of a few seconds, while others have half-lives of millions of years. The widespread use of 'exponential' to mean 'really fast' is a mistake, and dilutes the meaning of an important term.

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Ex`po*nen"tial (?), a. [Cf. F. exponentiel.]

Pertaining to exponents; involving variable exponents; as, an exponential expression; exponential calculus; an exponential function.

Exponential curve, a curve whose nature is defined by means of an exponential equation. -- Exponential equation, an equation which contains an exponential quantity, or in which the unknown quantity enters as an exponent. -- Exponential quantity Math., a quantity whose exponent is unknown or variable, as ax. -- Exponential series, a series derived from the development of exponential equations or quantities.


© Webster 1913.

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