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

1. Jun 21, 2010

### 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

2. Jun 22, 2010

### 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.

3. Jun 22, 2010

### 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.

4. Jun 22, 2010

### TheDonk

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

5. Jun 22, 2010

### 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 doesnt seem to work?

6. Jun 22, 2010

### Mute

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?)

Last edited: Jun 22, 2010
7. Jun 22, 2010

### luma

8. Jun 23, 2010

### Gregg

I have managed to solve things in terms of the w function where mathematica cant e.g. 1/(x+1)^(x+1). So I agree that maybe it isnt that great at solving it. But still, this particular function doesnt seem to work and i cant imagine a substitution that will do it