Question

How do you find the initial population in an exponential growth model?

Answer 1

Hint: We are given that the model is exponential, hence we need to know some of the exponential and logarithmic properties. Before that we should also know $$\ln$$ is a natural logarithm with its base always equals to $$e$$. The first logarithmic property we should know states that, $$\ln \left(a^b\right)=b\ln (a)$$. The second property states that if the base and argument of a logarithm are the same then its value is 1, by using this we can say that $$\ln (e)=1$$.

Complete step-by-step solution:
Let’s say that the population growth model is $$P(t)=P_0{e}^{kt}$$. In this equation $$P(t)$$ is the population after t years, k is the exponential constant, t is the number of years, and $$P_0$$ is the initial population.
We want to find the initial population, which means that we want the $$P(t)=P_0$$. Assume that we get $$P(t)=P_0$$ at $$t=t$$.
By substituting these values in the population growth model, we get
$$\begin{array}{rl}& \Rightarrow P(t)=P_0{e}^{kt}\\ & \Rightarrow P_0=P_0{e}^{kt}\end{array}$$
Dividing both sides of the above equation by $$P_0$$, we get
$$\Rightarrow {\displaystyle \frac{P_0}{P_0}}={\displaystyle \frac{P_0{e}^{kt}}{P_0}}$$
$$\Rightarrow 1=e^{kt}$$
Flipping the above equation, we get
$$\Rightarrow e^{kt}=1$$
Taking $$\ln$$ of both sides of the above equation, we get
$$\Rightarrow \ln \left(e^{kt}\right)=\ln (1)$$
Using the logarithmic property, $$\ln \left(a^b\right)=b\ln (a)$$. And the value of $$\ln (1)$$ equals zero, in the above equation we get
$$\Rightarrow kt\ln (e)=0$$
As the base and argument of $$\ln (e)$$ is the same, its value equals $$1$$.
$$\Rightarrow kt=0$$
$$k$$ is a non-zero exponential constant, hence $$t$$ must be zero for the above equation to be true.
It means that we can get the initial population by substituting $$t=0$$ in the population growth model.

Note: This is not only applicable to an exponential model, for any model be it polynomial, logarithmic, trigonometric, etc. If the model is showing growth or decay of a certain entity, then to find the initial amount we need to evaluate $$f(0)$$. Where $$f$$ growth/ decay model equation.