Expert: Sherman D. - 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
if you are saying you want me to solve for "x", then ok.

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

this becomes

y = a(1 + e)^((x - c)/b), because of the negative exponent in the denominator.

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

ln both sides and you get

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

divide both sides by ln(1 + e)

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

then the rest is easy

x = b(ln(y/a)/ln(1 + e)) + c or (bln(y/a) + cln(1 + e))/ln(1 + e) or (bln(y) - bln(a) + cln(1 + e))/ln(1 + e), just in case they have theirs different.

if you go to www.quickmath.com and put in your equation under Solve which is under Equation, it will give you the same thing i gave you, only they will put it as one fraction, also they use log instead of ln but they would would give you the same value in this case.

by the way, you may or may not know, but ln is the inverse of e.

if this is wrong, then just let me know what the answer is suppose to be.