Expert: Steve Holleran - 8/21/2008

Question
Hi
I am having a problem solving this equation for x:
y =   a
  ----------
      -((x-c)/b)
  1+ e

I tried to type it in the best I could. It is a sigmoidal equation. I have solved it a couple of differnet ways however I am not getting the correct answers for a known y and x values. I am using the equation to solve for unknown x values. Hopefully you can help me. Thanks!

Answer
Hi Kim,

Let's see if I can write this clearly enough.  Okay, we start with

            y = a / [1 + e^-( (x-c)/b)

So we can swap the y and the right side denominator:

       1 + e^-( (x-c) / b) = a / y

           e^-( (x-c) / b) = (a / y) - 1 = (a - y) / y


         1 / [e^( (x-c) / b) = (a - y) / y

so            e^( (x-c) / b ) = y / (a - y)

now changing to a logarithm expression:

             ln[y / (a-y)] = (x-c) / b

           b * ln[y / (a - y)] = x - c

and then    x = b * ln[y / (a-y)] + c


I hope this is the result you were looking for.
Steve