[Math] How to solve the equation $x \log \log x = n$

logarithms

I would like to solve the equation $x \log\log x = n$. I've seen a lot of post about the equation $x \log x$ but here I have a composition of $\log$.
How can I solve it ?

Thank you very much.

Best Answer

With the estimate $x=\dfrac n{\log(\log(n))}$, you have

$$\dfrac n{\log(\log(n))}\log\left(\log\left(\dfrac n{\log(\log(n))}\right)\right)=\dfrac n{\log(\log(n))}\log\left(\log(n)-\log\left(\log(\log(n))\right)\right)$$

which is asymptotically $n$.

For example, for $n=100$,

$$x=\frac{100}{\log(\log(100))}=65.4801821\cdots$$

and

$$x\ln(\ln(x))=93.684410\cdots.$$

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