I have a question about the Logistic growth model, I am applying the equation $$ y = \frac{\epsilon_1}{1 + e^{-(\epsilon_2+\epsilon_3x)}} $$ in a non linear regression.
I am following a book that first tried to convert this into a linear model, so this is my attempt:
$$y \approx \frac{\epsilon_1}{1 + e^{-(\epsilon_2+\epsilon_3x)}} \Rightarrow \frac{y}{\epsilon_1} \approx \frac{1}{1 + e^{-(\epsilon_2+\epsilon_3x)}} \Rightarrow \ ?¿$$
Here I am stuck because I do not know how to continue, I tried to apply the logit transformation $log \left[ \frac{r}{1-r} \right]$ with $r = \frac{y}{\epsilon_1}$. But I couldn't follow how, I am a biologist so it is a little bit complicated for me heheh
Could someone give me a hint to continue this, I should get that $$log\left[ \frac{\frac{y}{\epsilon_1}}{1-\frac{y}{\epsilon_1}}\right] \approx \epsilon_2 + \epsilon_3 x$$
Thank you in advance for your time and help.
$$y \approx \frac{\epsilon_1}{1 + e^{-(\epsilon_2+\epsilon_3x)}} \\ \Rightarrow \frac{y}{\epsilon_1} \approx \frac{1}{1 + e^{-(\epsilon_2+\epsilon_3x)}} \\ \Rightarrow 1-\frac{y}{\epsilon_1} \approx \frac{e^{-(\epsilon_2+\epsilon_3x)}}{1 + e^{-(\epsilon_2+\epsilon_3x)}} \\ \Rightarrow \frac{\frac{y}{\epsilon_1}}{1-\frac{y}{\epsilon_1}} \approx \frac{1}{e^{-(\epsilon_2+\epsilon_3x)}} = e^{(\epsilon_2+\epsilon_3x)} \\ \Rightarrow \log_e\left[\frac{\frac{y}{\epsilon_1}}{1-\frac{y}{\epsilon_1}}\right] \approx {\epsilon_2+\epsilon_3x}$$