Question

How to calculate the exponential growth factor?

Answer 1

Hint: The exponential growth factor represents the exponential increase in the growth of a quantity with respect to time; we will see the steps and the example of finding the exponential growth factor. Finally we conclude the required answer.

Complete step-by-step solution:
The exponential growth can be calculated using the following steps:
Step $1$: We have to determine the initial or the present value for the quantity we want to find the future value.
Step $2$: Try to determine the growth rate of that quantity, it can be in terms of years or months to even minutes and seconds.
Step $3$: now the tenure or the total period for which the quantity will stay in the system has to be figured out.
Step $4$: now the total number of periods in the tenure have to be calculated for which the growth has to be compounded and calculated.
Step $5$: Finally, the exponential growth will be calculated by compounding the initial value, the annual growth rate and the number of periods in the tenure.
Now let’s consider an example of a doubling life of a quantity:
Now exponential growth is generally represented by the formula: $P(t) = {P_0}{e^{kt}}$.
Where $P(t)$ is the value of the quantity at time $t$ and $k$ is the exponential growth factor and ${P_0}$ is the initial value.
Now to find the value of $k$, we need another equation given.
Let’s consider the doubling life of the quantity be $3$ years, therefore:
 ${P_0}{e^{k(3)}} = 2{P_0}$.
Now on cancelling ${P_o}$ from both the sides we get:
${e^{3k}} = 2$
On taking log to the base $e$ on both the sides, we get:
$\ln {e^{3k}} = \ln 2$
Now on simplifying we get:
$3k\ln e = \ln 2$
Now we know that $\ln e = 1$, therefore we get:
$3k = \ln 2$
Thus, $k = \dfrac{{\ln 2}}{3}$
Now on using a scientific calculator we get:
$k = 0.2310$

Which is the value of $k$ which is the exponential growth factor or the constant.

Note: Exponential growth and decay is used in the science fields to predict what will be the state of a quantity in the future, a quantity would increase which is called growth, or a quantity can decrease which is called decay.