Expert: Socrates - 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
take the reciprocal of both sides and get

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

multiply both sides by a

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

subtract 1 from both sides

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

take the natural log of both sides


ln (a/y - 1) = -((x-c)/b))

multiply both sides by -b

-b ln (a/y - 1) = x-c

add c to both sides

c-bln(a/y - 1) = x

x = c-bln(a/y - 1)

if you prefer , you can rewrite this as

x = c+ln(y/(a-y))^b