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 Apr 22 comment Approximation of a quotient that involves the Lambert function. I understood that there are two solutions, and, in fact, that disturbed me a little. Thanks again for your answer :-) Apr 22 accepted Approximation of a quotient that involves the Lambert function. Apr 21 comment Approximation of a quotient that involves the Lambert function. I want to solve the equation $xn^{1/x}=(\ln n)^{1+c}$ with an asymptotic approximation. I had the intuition that $x=O(\ln n/(\ln \ln n))$. Apr 21 comment Approximation of a quotient that involves the Lambert function. Thanks very much for your detailed answer @JJacquelin. So, what branch should I use ? $W_{-1}$ or $W_0$ ? Apr 21 comment Approximation of a quotient that involves the Lambert function. Thanks again :-) Apr 21 comment Approximation of a quotient that involves the Lambert function. I don't understand very well. You say that I have to choose what I want ? Apr 21 comment Approximation of a quotient that involves the Lambert function. Thanks very much @user90369. If have to use $W_{-1}$, then we have $\frac{-\ln n}{W_{-1}(- \ln^{-c}n)}=O\left(\frac{\ln n}{\ln \ln n}\right)$. What is the argument for using $W_{-1}$ ? Thanks again. Apr 21 comment Approximation of a quotient that involves the Lambert function. Thanks you @user90369. Your proof is interesting. Is $k^{1/k}$ always lower than $e^{1/e}$ ? I don't know this property. Using the Lambert function allows to obtain an accurate solution (with a denominator which is function of $n$). It is true that I'm interested in a solution $x < \ln n$; but I would prefer a tight asymptotic upper bound. To do that, I think we have to use the branch $W_{-1}$, but I don't know if we can. Thanks again. Apr 20 comment Approximation of a quotient that involves the Lambert function. In fact I'm looking for a slightly tighter bound. For information: I've obtained $x=\frac{-\ln n}{W(- \ln^{-c}n)}$ by solving the equation $xn^{1/x}= (\ln n)^{1+c}$. I know that $x$ should lower than $\ln n$, but I don't know in what extent... Apr 20 comment Approximation of a quotient that involves the Lambert function. Thanks for your answer. What about the second branch $W_{-1}$ ? I'm reading on Wiki that when $x$ approaches $0$, we have $W_{-1}(x)$ close to $\ln(-x) - \ln(-\ln(-x))$. Apr 20 asked Approximation of a quotient that involves the Lambert function. Apr 20 comment Problem of simplification Thanks for your comments. Apr 20 comment Problem of simplification Thanks very much. Apr 16 comment Problem of simplification Thanks for your comment @martycohen. Do you want to say that I should be satisfied with such a form ? Apr 16 comment Problem of simplification @ClaudeLeibovici In fact, I don't like the negative signs. Apr 16 comment Problem of simplification @ClaudeLeibovici I've added it. Apr 16 revised Problem of simplification added 34 characters in body Apr 16 asked Problem of simplification Apr 8 comment How to solve the equation $x \log \log x = n$ Thanks for this comment. Apr 8 accepted How to solve the equation $x \log \log x = n$