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