Exponential Growth and Decay

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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 real-world 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 _oe ^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.