Let $L=\log(1/\epsilon)$. Taking logarithms gives $x^2+\log x=L$. Equivalently, $2x^2=W(2/\epsilon^2)$ in terms of the <Lambert W function>. Iteration for large $L$ gives $x^2=L-\tfrac12\log L+O((\log L)/L)$ and hence
$$
\boxed{x=\sqrt L-\frac{\log L}{4\sqrt L}
+O\left(\frac{(\log L)^2}{L^{3/2}}\right).}
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 336 / 1 / a / Solution

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The Lambert W function is the multivalued inverse of $w\mapsto we^w$, so $W(z)e^{W(z)}=z$. Its principal real branch satisfies $W(z)=\log z-\log\log z+o(1)$ as $z\to+\infty$.


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