Y = 1/(1+x)^x rearranged in terms of x

[TheDonk]

edit: Oh nuts, embarrassing typo. Should be "...rearranged in terms of y"
Ok, I swear this isn't homework but if it seems too much like homework I'm happy to have it moved there.

So I need to be able to find x given y in:

y = 1/(1+x)^x

Possibly important:
y will always be between 0 and 1 and as y goes from 0 to 1, x goes from infinity to 0.

I don't really know where to start. Is there an exact algebraic solution at all? I need to write an algorithm that will calculate this reliably. The reason I need it is to calculate a sigmoid-like curve based on certain (in my opinion) intuitive inputs. The one remaining problem is inverting this equation!

Thanks to all who help!
- Andrew

 

[Mute]

There is no algebraic way to do it, I expect. You may be able to express it in terms of the Lambert W function (http://en.wikipedia.org/wiki/Lambert_W_function). I played around with it quickly but didn't immediately figure out how to manipulate it to get it into Lambert W form.

 

[Tedjn]

I wonder whether Newton's method is a good numerical approach to this. Or there may be libraries out there that can do this sort of thing -- I don't know.

 

[TheDonk]

That Lambert W function will definitely give me something to think about. Thanks!

 

[Gregg]

That function will not work in this case. you can easily do 1/x^x or 1/(x+1)^{x+1} but this one doesn't seem to work?

 

[Mute]

That function will not work in this case. you can easily do 1/x^x or 1/(x+1)^{x+1} but this one doesn't seem to work?

It's true that Mathematica doesn't seem to be able to find a solution in terms of the Lambert W function (known in Mathematica as ProductLog). I was thinking perhaps a substitution could be found to get the thing into productlog form, but perhaps there isn't one (or Mathematica isn't good at solving for things in terms of the productlog if it doesn't have a built in answer?)

 

[luma]

if you need a sigmoid like curve then why not use 1 / (1 + e^-x) ?

http://en.wikipedia.org/wiki/Logistic_function

It is a very nice function to use.

 

[Gregg]

It's true that Mathematica doesn't seem to be able to find a solution in terms of the Lambert W function (known in Mathematica as ProductLog). I was thinking perhaps a substitution could be found to get the thing into productlog form, but perhaps there isn't one (or Mathematica isn't good at solving for things in terms of the productlog if it doesn't have a built in answer?)

I have managed to solve things in terms of the w function where mathematica can't e.g. 1/(x+1)^(x+1). So I agree that maybe it isn't that great at solving it. But still, this particular function doesn't seem to work and i can't imagine a substitution that will do it