Consider the Gompertz equation that models the dynamics of the population of a single-species $$\frac{dN}{dt}=r_0e^{-\alpha t}N$$ and convert it to the following form $$\frac{dN}{dt}=\alpha N\ln\left(\frac{K}{N}\right)$$
So here's my attempt, the problem doesn't explicitly say that the constant of integration is $0$ or not, so I consider two cases when solving and derived,
$$ N(t) = N_0 \exp \left( \frac{r_0}{\alpha}\left(1-e^{-\alpha t}\right) \right) $$
where I took $N(0)=N_0$, and, $N(t)=e^{-\frac{r_0}{\alpha}e^{-\alpha t}}$, for the case where the constant is $0$. So, taking the limit we arrive at the carrying capacity to get,
$$ K=\lim N=\lim e^{-\frac{r_0}{\alpha}e^{-\alpha t}}=e^{-\frac{r_0}{\alpha}}\iff\ln K=-\frac{r_0}{\alpha}\iff r_0=-\alpha\ln K $$
Now, observe
$$ N(t)=e^{-\frac{r_0}{\alpha}e^{-\alpha t}}\iff\ln N=-\frac{r_0}{\alpha}e^{-\alpha t}\Rightarrow\ln N=\ln Ke^{-\alpha t} $$
so that $\frac{\ln N}{\ln K}=e^{-\alpha t}$. Plugging all these in to the initial Gompertz equation should get you to arrive at the conclusion, I would believe. Now here's where I'm running into trouble, the fact that I have $\frac{\ln N}{\ln K}$ and not $\ln(\frac{K}{N})$. Not sure if it's here where my mistake lies, but if you can provide some guidance that would be useful. I'll attempt derivation using the first form of the solution I got without a constant of integration equaling to zero in the mean time.