How is the constant $k$ in $y = y_0\cdot e^{kt}$ equal to $\ln(1+z)$?

Question

The task is about population growth

and I have been given that you can find that the time when the population will be double is given at

$dt = \ln(2)/\ln(1+z)$ (Where $z$ is the growth rate)

So I have thought about this and the conclusion would be that the k value in the equation "$y = y_0\cdot e^{kt}$" must be $\ln(1+z)$. But I can't figure out how the $k$ value is $\ln(1+z)$.

I have also been given this equation if it helps (2518269)*(1+z).^(2015-1950) = 7223487, but I cannot seem to understand how that would help med understand k being log(1+z)

Thank you very much.

Answer 1

It seems not directly derived from the equation, indeed we have that

$$y=y_0\cdot e^{kt}=2y_0 \implies e^{kt}=2 \implies kt=\log 2 \implies t=\frac{\ln 2}k$$

with $k=\log (1+z)$.