What is "exponential growth"

Exponential \Ex`po*nen"tial\, a. [Cf. F. exponentiel.]

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

2. changing over time in an exponential manner, i. e. increasing or decreasing by a fixed ratio for each unit of time; as, exponential growth; exponential decay.

   Note:

   Exponential growth is characteristic of bacteria and other living populations in circumstances where the conditions of growth are favorable, and all required nutrients are plentiful. For example, the bacterium Escherichia coli in rich media may double in number every 20 minutes until one of the nutrients becomes exhausted or waste products begin to inhibit growth. Many fascinating thought experiments are proposed on the theme of exponential growth. One may calculate, for example how long it would take the progeny of one Escherichia coli to equal the mass of the known universe if it multiplied unimpeded at such a rate. The answer, assuming the equivalent of 10^ 80 hydrogen atoms in the universe, is less than three days. Exponential increases in a quantity can be surprising, and this principle is often used by banks to make investment at a certain rate of interest seem to be very profitable over time.

   Exponential decay is exhibited by decay of radioactive materials and some chemical reactions (first order reactions), in which one-half of the initial quantity of radioactive element (or chemical substance) is lost for each lapse of a characteristic time called the half-life.

   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 a^ x.

   Exponential series, a series derived from the development of exponential equations or quantities.

n. 1 The growth in the value of a quantity, in which the rate of growth is proportional to the instantaneous value of the quantity; for example, when the value has doubled, the rate of increase will also have doubled. The rate may be positive or negative. 2 (context by extension proscribed English) Very rapid growth.

Exponential growth is a phenomenon that occurs when the growth rate of the value of a mathematical function is proportional to the function's current value, resulting in its growth with time being an exponential function. Exponential decay occurs in the same way when the growth rate is negative. In the case of a discrete domain of definition with equal intervals, it is also called geometric growth or geometric decay, the function values forming a geometric progression. In either exponential growth or exponential decaay, the ratio of the rate of change of the quantity to its current size remains constant over time.

The formula for exponential growth of a variable x at the (positive or negative) growth rate r, as time t goes on in discrete intervals (that is, at integer times 0, 1, 2, 3, ...), is

x = x(1 + r)

where x is the value of x at time 0. For example, with a growth rate of r = 5% = 0.05, going from any integer value of time to the next integer causes x at the second time to be 5% larger than it was at the previous time. Since the time variable, which is the input to this function, occurs as the exponent, this is an exponential function.