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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.

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We have already explored some basic applications of exponential and logarithmic functions.

The formula is derived as follows \[\begin 20&= 10e^\ 2&= e^k \qquad \text\ \ln2&= k \qquad \text \end\] so \(k=\ln(2)\).

We observe that the coefficient of \(t\), \(\dfrac≈−1.2097×10^\) is negative, as expected in the case of exponential decay.In this section, we explore some important applications in more depth, including radioactive isotopes and Newton’s Law of Cooling.In real-world applications, we need to model the behavior of a function.In our choice of a function to serve as a mathematical model, we often use data points gathered by careful observation and measurement to construct points on a graph and hope we can recognize the shape of the graph.Exponential growth and decay graphs have a distinctive shape, as we can see in Figure \(\Page Index\) and Figure \(\Page Index\).