I have a logistic growth curve as follows:
$y = \frac{1}{(1 + ae^{-bx})}$, where x is the independent measure (x-axis) and a and b are paramaters. The inflection point of this equation is when y = 0.5.
Given that information, I need to find the x-value at y = 0.5. But this equation isn't easy to solve. I'm using the statistical language R to find these fits.
So, given the parameters a and b, how can I find the x-value at y = 0.5 (and again, I can't do things like derivatives because I am using a programming language). By the way, I did try to solve for x, but it's quite ugly and I'm not sure how to do it…
Best Answer
Solving for $x$, you get:
$$x = -\frac{1}{b} \ln\left(\frac{1-y}{ay}\right)$$
Insert $y = 0.5$ and you get:
$$x = -\frac{1}{b} \ln\left(\frac{1}{a}\right)$$
It isn't that ugly... Practice the math - I promise it will help you.