Logistic differential equation intrepetation

Question

If we have a logistic model of a function, given then the differential equation $$N'=k\cdot N\cdot (1-\frac{N}{B})$$ where $k$ is a constant of proportionality, $N$ is for example the function of the population and $B$ is the limit (or the capacity). The solution to this diff equation is $$N = B\cdot N_{0} \frac{e^{kt}}{B + N_{0} e^{kt} - N_{0}}$$ if we consider $N_0$ to be the initial value at $t = 0$. I am trying to make a connection between this solution and an ordinary logistic function, such as $$N(t)=\frac{C}{1+a\cdot e^{-bt}}$$ I've read that these two are supposed to be one and the same, but I need to show that. The method I followed was that I set $N(0)$ in the second function, so I obtained that $N_0 = \frac{C}{1+a}$, thus we have one constant interpreted. We also know the limits of both functions, namely $B=C$. But I am having trouble interpreting $a$ and $k$ here, so I get the wrong solution plugging those values in. Any ideas as to how to conclude this and show their resemblance?