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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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    $\begingroup$ This is what it usually takes to handle such things: 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... :( $\endgroup$
    – fedja
    Dec 13, 2012 at 14:22
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    $\begingroup$ 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. $\endgroup$
    – GH from MO
    Dec 13, 2012 at 14:24
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    $\begingroup$ 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. $\endgroup$
    – Henry Cohn
    Dec 13, 2012 at 14:59
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    $\begingroup$ 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. $\endgroup$
    – Alexandre Eremenko
    Dec 13, 2012 at 15:31
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    $\begingroup$ 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. $\endgroup$
    – B H
    Dec 13, 2012 at 15:53

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