I'd be interested in learning more about an apparent asymptotic relation to the natural logarithm which is provided in the second answer to this posted question:
Is there an approximation to the natural log function at large values?
Specifically, I'm seeking any results, proofs, discussions, history or references to the following relation: $$\ln \left( x \right) \approx a{x^{\frac{1}{a}}} – a$$
where $a$ is any constant and the larger we choose $a$ the better is the approximation to the natural log of $x$.
I've searched various pre-calc, calculus, diff eq and real analysis textbooks for discussions, or even something in the exercises, without success. I've also looked through two books on asymptotics, one on integrals and the other on real analysis, and there is nothing about this relation in either. The author of the second answer to the linked post has mentioned it was their own derivation, which is admirable. But I'm looking for anything and everything to understand this asymptotic relation better, including rates of convergence of the expression to $log(x)$.
Best Answer
This approximation is much less useful than it appears, because to calculate $x^{\frac{1}{a}}$ for large $a$ you'd likely write it as $\exp \left( \frac{\ln x}{a} \right)$ so you'd likely need to calculate $\ln x$ anyway.
Anyway, this is most of the way to a proof. For $a$ large (more specifically, as we are about to see, large compared to $(\ln x)^2$) we have
$$ax^{\frac{1}{a}} = a \exp \left( \frac{\ln x}{a} \right) = a \left( 1 + \frac{\ln x}{a} + O \left( \frac{(\ln x)^2}{a^2} \right) \right) = a + \ln x + O \left( \frac{(\ln x)^2}{a} \right)$$
using the first few terms of the Taylor series of $\exp(x)$. If you'd like you can write down a more precise error term using a form of Taylor's theorem with remainder.