Exponential Growth and Decay

In real-world applications, we need to model the behavior of a function. In mathematical modeling, we choose a familiar general function with properties that suggest that it will model the real-world phenomenon we wish to analyze.

In the case of rapid growth, we may choose the exponential growth function, [latex]A={A}_{0}{e}^{rt}[/latex], where [latex]{A}_{0}[/latex] (used to be labeled as [latex]P[/latex]) is equal to the value at time zero, [latex]e[/latex] is Euler’s constant, and [latex]r[/latex] is a positive constant that determines the rate (percentage) of growth.

We may use the exponential growth function in applications involving doubling time, the time it takes for a quantity to double. Such phenomena as wildlife populations, financial investments, biological samples, and natural resources may exhibit growth based on a doubling time. In some applications, however, as we will see when we discuss the logistic equation, the logistic model sometimes fits the data better than the exponential model.

On the other hand, if a quantity is falling rapidly toward zero, without ever reaching zero, then we should probably choose the exponential decay model. Again, we have the form [latex]y={A}_{0}{e}^{-kt}[/latex] where [latex]{A}_{0}[/latex] is the starting value, and [latex]e[/latex] is Euler’s constant. Now [latex]k[/latex] is a negative constant that determines the rate of decay. We may use the exponential decay model when we are calculating half-life, or the time it takes for a substance to exponentially decay to half of its original quantity. We use half-life in applications involving radioactive isotopes.

A population of bacteria doubles every hour. If the culture started with [latex]10[/latex] bacteria, graph the population as a function of time.

Show Solution

When an amount grows at a fixed percent per unit time, the growth is exponential. To find [latex]{A}_{0}[/latex] we use the fact that [latex]{A}_{0}[/latex] is the amount at time zero, so [latex]{A}_{0}=10[/latex]. To find [latex]r[/latex], use the fact that after one hour [latex]\left(t=1\right)[/latex] the population doubles from [latex]10[/latex] to [latex]20[/latex]. The formula is derived as follows:

[latex]\begin{array}{l}\text{ }20=10{e}^{r\cdot 1}\hfill & \hfill \\ \text{ }2={e}^{r}\hfill & \text{Divide both sides by 10}\hfill \\ \mathrm{ln}2=r\hfill & \text{Take the natural logarithm of both sides}\hfill \end{array}[/latex]

so [latex]r=\mathrm{ln}\left(2\right)[/latex]. Thus the equation we want to graph is [latex]y=10{e}^{\left(\mathrm{ln}2\right)t}=10{\left({e}^{\mathrm{ln}2}\right)}^{t}=10\cdot {2}^{t}[/latex]. The graph is shown below.

Analysis of the Solution
[latex]\\[/latex]
The population of bacteria after ten hours is [latex]10,240[/latex]. We could describe this amount as being of the order of magnitude [latex]{10}^{4}[/latex]. The population of bacteria after twenty hours is 10,485,760 which is of the order of magnitude [latex]{10}^{7}[/latex], so we could say that the population has increased by three orders of magnitude in ten hours.

Calculating Doubling Time

For growing quantities, we might want to find out how long it takes for a quantity to double. As we mentioned above, the time it takes for a quantity to double is called the doubling time.

Given the basic exponential growth equation [latex]A={A}_{0}{e}^{kt}[/latex], doubling time can be found by solving for when the original quantity has doubled, that is, by solving [latex]2{A}_{0}={A}_{0}{e}^{kt}[/latex].

The formula is derived as follows:

[latex]\begin{array}{l}2{A}_{0}={A}_{0}{e}^{kt}\hfill & \hfill \\ 2={e}^{kt}\hfill & \text{Divide both sides by }{A}_{0}.\hfill \\ \mathrm{ln}2=kt\hfill & \text{Take the natural logarithm of both sides}.\hfill \\ t=\frac{\mathrm{ln}2}{k}\hfill & \text{Divide by the coefficient of }t.\hfill \end{array}[/latex]

Thus the doubling time is

[latex]t=\frac{\mathrm{ln}2}{k}[/latex]