# Thread: Stuck with finding roots including population models

1. ## Stuck with finding roots including population models

Hi,

I would like to find the solution for t in;

a*t=1/(1+c*exp(-b*t))

where a, b and c are all positive and less than 1. Left hand side of the function is cumulative logistics curve. I know it is not invertible, I am assuming the solution would include LambertW function. Is there a way to find t in closed form (using LambertW or any other functions)?
Thanks,
rutabaga

2. Originally Posted by rutabaga
. . . or any other functions)?
Thanks,
rutabaga
Here's a numerical approach in Mathematica:

Code:
f[a_, b_, c_, t_] := a^t - 1/(1 + c/E^(b*t));
g[x_, y_, z_] := t /. FindRoot[f[x, y, z, t] == 0, {t, 0}]