Exponential extrapolation of Earth's population

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In 2025 the Earth’s population is about $10$ billion; around 1800 it was about $1$ billion. Assuming exponential growth $f(t)=a\cdot b^{t}$ (with $t$ = year on the Gregorian calendar), determine in which year the first human would have appeared on Earth.

Solution

From $\dfrac{f(2025)}{f(1800)}=b^{225}=10\Rightarrow b=10^{1/225}$. “First human” means a single individual, $f(t)=1$. Using the ratio: $b^{\,t-1800}=\dfrac{1}{10^{9}}=10^{-9}=b^{-9\cdot225}=b^{-2025}$ (since $10=b^{225}$), hence $t-1800=-2025\Rightarrow t=-225$. The model places the “first human” in the year $-225$ (225 BCE): a plainly absurd result, which exposes the limits of extrapolating exponential growth backward in time.

$\boxed{t\approx-225\ (\text{year }225\text{ BCE})}$