The law of uninhibited growth is express as:
[tex]N(t)=N_0e^{kt}[/tex]
where N(t) is the population at time t, N0 is the initial population, k is the growth rate and t is the time. In this case we know that after one day, t=1, the population is 800 and that the initial population was 500; plugging these values and solving for k we have:
[tex]\begin{gathered} 800=500e^k \\ e^k=\frac{800}{500} \\ \ln e^k=\ln\frac{8}{5} \\ k=\ln\frac{8}{5} \end{gathered}[/tex]
Now that we have the growth rate, we know that the population growth in this case can be express as:
[tex]N(t)=500e^{(\ln\frac{8}{5})t}[/tex]
We want to know the time it takes for the population to be 7000, to find it we equate our expression to this value and solve for t:
[tex]\begin{gathered} 7000=500e^{(\ln\frac{8}{5})t} \\ e^{(\ln\frac{8}{5})t}=\frac{7000}{500} \\ \ln e^{(\ln\frac{8}{5})t}=\ln14 \\ (\ln\frac{8}{5})t=\ln14 \\ t=\frac{\ln14}{\ln\frac{8}{5}} \\ t=5.6 \end{gathered}[/tex]
Therefore, it takes 5.6 days for the population to reach 7000 individuals.
