If $2^x/\sqrt{x}=n$, how does $x$ grow asymptotically?

Question

If $\dfrac{2^x}{\sqrt{x}}=n$ where $x\rightarrow\infty$ as $n\rightarrow\infty$, how does $x$ grow asymptotically in terms of $n$?

We know that it grows faster than $\log n$, because $\dfrac{2^{\log n}}{\sqrt{\log n}}=\dfrac{n}{\sqrt{\log n}}<n$. But it grows slower than $(\log n)^2$, because $\dfrac{2^{(\log n)^2}}{\sqrt{(\log n)^2}}=\dfrac{n^{\log n}}{\log n}>n$.

Answer 1

Taking logs (base 2) gives

$$ \lg n = x - \frac{\lg x}{2}, \tag{1} $$

and, dividing through by $x$,

$$ \frac{\lg n}{x} = 1 - \frac{\lg x}{2x}. $$

We observe that $x \to \infty$ as $n \to \infty$, so $(\lg x)/(2x) \to 0$ as $n \to \infty$, and thus

$$ \lim_{n \to \infty} \frac{\lg n}{x} = 1. $$

Equivalently,

$$ x \sim \lg n \qquad \text{as } n \to \infty. $$

Answer 2

Working in log to the base 2, let $y=x-\log n$ so that $y=\frac{\log x}{2}$; using that $\log n \leq x \leq 2 \log n$ (the latter bound can be applied by one more iteration of OP's argument), we get: $x-\log n - \frac{1}{2}\log \log n \in [0,c]$ for a small constant $c$.