Answer
$f(x)=\displaystyle \frac{20}{1+3(0.8)^{-x}}$
Work Step by Step
Logistic model: $f(x)=\displaystyle \frac{N}{1+Ab^{-x}}$,
where $N$ = limiting value.
While x is relatively small, $f$ behaves exponentially, $f(x)\displaystyle \approx(\frac{N}{1+A})b^{x}.$
We are given
$\left\{\begin{array}{l}
N=20,\\
b=100\%-20\%=0.8,\\
f(0)=5
\end{array}\right.$
We plug this in to obtain:
$f(0)=5\quad\Rightarrow\left\{\begin{array}{ll}
5 & =\dfrac{20}{1+Ab^{0}}\\
1+A & =\dfrac{20}{5}\\
A & =4-1=3
\end{array}\right.$
Thus we get:
$f(x)=\displaystyle \frac{20}{1+3(0.8)^{-x}}$
