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I asked this on MSE but no one could get it. https://math.stackexchange.com/questions/257455/inverse-function-of-y-frac-lnx1-ln-x It's been bothering me for a really long time. I think it's equivalent to finding the real positive root of $x^n-x-1$ for any real $n$ that isn't zero. Is it possible to do something like this? I think the Lambert W function might prove helpful, but I don't really know how to manipulate that well myself.

What is $f(x)$ if $f^{-1}(x)=\frac{\ln(x+1)}{\ln x}$?

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B H
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  • Should $f-1(x)$ by $f^{-1}(x)$? – Lee Mosher Dec 13 '12 at 13:53
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    This is what it usually takes to handle such things: http://www.artofproblemsolving.com/Forum/viewtopic.php?f=67&t=436858 I don't think I have Lambert covered there but the corresponding elementary transcendental extension with the rule $dt\frac{t+1}{t}=\frac {df}f$ is also manageable. Try it! I really have no time now and it is a 2-day project at the very least... :( – fedja Dec 13 '12 at 14:22
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    This is not a real question, because it is not clear what kind of answer you would accept. To me there is a simple answer: $f(x)$ is the inverse of the given fraction, period. – GH from MO Dec 13 '12 at 14:24
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    As GH says, it's not clear what constitutes an answer. Expressing this function in terms of the $W$ function is the only positive outcome that seems plausible to me, but that would just amount to identifying it as being essentially the same as another problem with no standard closed form. I don't have any intution for what can be expressed in terms of $W$ and what can't, and while I don't care much about this particular function, any intuition for the general question could be interesting. – Henry Cohn Dec 13 '12 at 14:59
  • I hadn't looked carefully at Fedja's comment before I wrote mine. Someone should really take up his suggestion (I wish I had time). – Henry Cohn Dec 13 '12 at 15:02
  • I agree with GH. You have to say what kind of solution you want. Usually one solves such equations in the form of power series. But you can expand at different points. Function $f^{-1}$ is defined for positive $x$ and has a simple pole at the point $x=1$. Where do you want a power series expnsion? – Alexandre Eremenko Dec 13 '12 at 15:25
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    If I understand correctly Fedja's suggestion, he proposes to prove that $f$ is not an elementary function (whatever this means). But I am not sure that this is what you want. So you must state your question precisely. – Alexandre Eremenko Dec 13 '12 at 15:31
  • By the way, my interpretation was that Fedja was proposing to prove not just that it is not elementary, but also that using $W$ doesn't help (or at least to figure out whether it does). – Henry Cohn Dec 13 '12 at 15:33
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    Sorry for the radio silence. I like this function because it'd return interesting values for integer arguments, like the golden ratio for 2 and the plastic number for 3. I was hoping for a solution over the reals; that is, the inverse of $f$ when $x>0$. I'm looking for something in terms of transcendental functions like $W$ rather than a sum or an integral or anything like that. – B H Dec 13 '12 at 15:53
  • Also, thank you, fejda, yours seems like an excellent answer, but it's way beyond me. – B H Dec 13 '12 at 16:03
  • You are welcome, Benjamin! If what I wrote on AoPS is currently slightly above your head, another suggestion is to push this question to the back of your memory for now and to return to it right after you have finished your algebra courses in college (or after you've read some algebra textbook like van der Waerden to the middle or so :)). – fedja Dec 13 '12 at 17:11
  • @fedja Your AoPS link has a 404 error – Tyma Gaidash Nov 26 '22 at 17:45
  • related answers: https://mathoverflow.net/questions/426543/is-it-possible-to-solve-for-y-in-this-equation https://math.stackexchange.com/questions/257455/inverse-function-of-y-frac-lnx1-ln-x https://math.stackexchange.com/questions/4491579/is-it-possible-to-solve-the-equation-x-1-x-y-explicitly/4492364#4492364 – IV_ Dec 02 '22 at 19:30

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