Question

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

Answer 1

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

& \Rightarrow P(t)={{P}_{0}}{{e}^{kt}} \\\ & \Rightarrow {{P}_{0}}={{P}_{0}}{{e}^{kt}} \\\ \end{aligned}$$ Dividing both sides of the above equation by $${{P}_{0}}$$, we get $$\Rightarrow \dfrac{{{P}_{0}}}{{{P}_{0}}}=\dfrac{{{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.