Let's do the usual thing and make t the years since 1950. We'll just abbreviate a billion B.
f(1950-1950)=39.4 B
f(2000-1950) =2025.2 B
Our exponential form for f will be
[tex]f = a e^{kt}[/tex]
[tex]39.4 \textrm{ B} = a e^{ 0 k} = a[/tex]
[tex]2025.2 \textrm{ B} = a e^{50 k}[/tex]
Dividing
[tex]\dfrac{2025.2}{39.4} = e^{50 k}[/tex]
[tex]50 k = \ln \dfrac{2025.2}{39.4}[/tex]
[tex]k = \frac 1 {50} \ln \dfrac{2025.2}{39.4} \approx 0.0787932[/tex]
Our function is
[tex]f = 39.4 \textrm{ B } e^{0.0787932 t }[/tex]
Since [tex]e^{0.0787932} \approx 1.08198[/tex]
[tex]f = 39.4 \textrm{B } 1.08198^t }[/tex]
around 8.2 % annualized growth.
