Exponential Growth and Decay
Exponential Growth and Decay
An exponential function is a function that has a variable as an exponent and the base is positive and not equal to one. For example, f (x ) = 2^{x} is an exponential function. Note that f (x ) = x ^{2} is not an exponential function (but instead a basic polynomial function), because the exponent is a constant and not a variable. Exponential functions have graphs that are continuous curves and approach but never cross a horizontal asymptote . Many realworld
processes follow exponential functions or their inverses, logarithmic functions.
Exponential Change
Exponential growth is a mathematical change that increases without limit based on an exponential function. The change can be in the positive or negative direction. The important concept is that the rate of change continues to increase. Exponential decay is found in mathematical functions where the rate of change is decreasing and thus must reach a limit, which is the horizontal asymptote of an exponential function. In the figure above, the asymptote is the x axis where the rate of change approaches zero. Exponential decay may also be either decreasing or increasing; the important concept is that it progresses at a slower and slower rate.
Exponential growth and decay are modeled in many realworld processes. Populations of growing microbes, and indeed a growing population of any life when not constrained by environmental factors such as available space and nutrition, can be modeled as a function showing exponential growth. The growth of a savings account collecting compound interest is another example of an exponential growth function.
Exponential decay is seen in many processes as well. The decrease in radioactive material as it undergoes fission and decays into other atoms fits a curve of exponential decay. The discharge of an electric capacitor through a resistance can be calculated using exponential decay. A warm object as it cools to a constant surrounding temperature, or a cool object as it warms, will exhibit a curve showing exponential decay.
A Sample Problem
The following is an example of how the mathematics of exponential growth and decay can be used to solve problems. Suppose that a radioactive sample, measured after 2 days, had only 60 percent as much of the sample as it had initially. How much of the sample could be expected to remain after 5 days?
To solve this problem, it must be understood that the sample is reducing in size by exponential decay and the rate at which it is reducing must first be determined. Formulas for this exponential decay are as follows:
where N _{o} is the original value, N is the new value, k is decay constant or rate of change, and t is time. Using the second formula we see that N _{o} is 100 and N is 60, and t (in days) is 2, we get which is −0.5108 = k 2 or −0.2554 = k. So the decay rate is −0.2554. Now that we know the decay rate, to find out what happens in 5 days we use the first formula N = N _{o}e ^{kt} where N = 100e ^{5(−0.2554)} and so N = 27.89. Thus we will have 27.89 percent of the sample after 5 days.
see also Logarithms; Powers and Exponents.
Harry J. Kuhman
Bibliography
Dykes, Joan, and Ronald Smith. Finite Mathematics with Calculus. New York: Harper Perennial, 1993.
Trivieri, Lawrence A. Precalculus Mathematics, Functions and Graphs. New York: Harper Perennial, 1993.
Cite this article
Pick a style below, and copy the text for your bibliography.

MLA

Chicago

APA
"Exponential Growth and Decay." Mathematics. . Encyclopedia.com. 17 Oct. 2018 <http://www.encyclopedia.com>.
"Exponential Growth and Decay." Mathematics. . Encyclopedia.com. (October 17, 2018). http://www.encyclopedia.com/education/newswireswhitepapersandbooks/exponentialgrowthanddecay
"Exponential Growth and Decay." Mathematics. . Retrieved October 17, 2018 from Encyclopedia.com: http://www.encyclopedia.com/education/newswireswhitepapersandbooks/exponentialgrowthanddecay
Citation styles
Encyclopedia.com gives you the ability to cite reference entries and articles according to common styles from the Modern Language Association (MLA), The Chicago Manual of Style, and the American Psychological Association (APA).
Within the “Cite this article” tool, pick a style to see how all available information looks when formatted according to that style. Then, copy and paste the text into your bibliography or works cited list.
Because each style has its own formatting nuances that evolve over time and not all information is available for every reference entry or article, Encyclopedia.com cannot guarantee each citation it generates. Therefore, it’s best to use Encyclopedia.com citations as a starting point before checking the style against your school or publication’s requirements and the mostrecent information available at these sites:
Modern Language Association
The Chicago Manual of Style
http://www.chicagomanualofstyle.org/tools_citationguide.html
American Psychological Association
Notes:
 Most online reference entries and articles do not have page numbers. Therefore, that information is unavailable for most Encyclopedia.com content. However, the date of retrieval is often important. Refer to each style’s convention regarding the best way to format page numbers and retrieval dates.
 In addition to the MLA, Chicago, and APA styles, your school, university, publication, or institution may have its own requirements for citations. Therefore, be sure to refer to those guidelines when editing your bibliography or works cited list.