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

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
\[\begin{align}
  & \Rightarrow P(t)={{P}_{0}}{{e}^{kt}} \\
 & \Rightarrow {{P}_{0}}={{P}_{0}}{{e}^{kt}} \\
\end{align}\]
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.