Question

How do you know when to use $ e $ in exponential growth models?

Answer 1

Hint: To solve this question, we need to use the concept of exponential growth. The exponential growth occurs when the growth rate of the value of a mathematical function is proportional to the function’s current value, resulting in its growth with time being an exponential function. We can also say that when the growth of a function increases rapidly in relation to the increase in the total number, then it is exponential.

Complete step by step solution:
We know that $ e $ can be used whenever systems grow exponentially and continuously: population, radioactive decay, interest calculations, and more. Even the systems that do not grow smoothly can be approximated by $ e $ .
To understand it in a better way, we can say that Just like every number can be considered a scaled version of 1 (the base unit), every circle can be considered a scaled version of the unit circle (radius 1), and every rate of growth can be considered a scaled version of $ e $ (unit growth, perfectly compounded).
Thus, we can say that $ e $ represents the idea that all continually growing systems are scaled versions of a common rate.
Also, we will have to use the logarithms in case of the exponential growth. It will be easy for us to use $ e $ at that time because it is easy to calculate and get the required answer.

Note: Systems that experiences exponential growth increase according to the mathematical model:
\[P = {P_0}{e^{kt}}\]
Where, $ {P_0} $ represents the initial state of the system and $ k $ is a constant, called the growth constant.