TITLE: Can $e^{\frac{1}{\ln x}}$ be simplified or roughly approximated
QUESTION [6 upvotes]: Clearly $e^{\ln x} = x$.  
Is there a simplification for $e^{\frac{1}{\ln x}}$

REPLY [2 votes]: We have that for $x$ large or small since $\frac1{\log x} \to 0$
$$e^{\frac{1}{\ln x}} \approx1+\frac1{\ln x}$$
and more in general
$$e^{\frac{1}{\ln x}} \approx 1+\frac1{\ln x}+\frac12 \frac1{\ln^2 x}+\frac1{3!}\frac1{\ln^3 x}+\ldots+\frac1{n!}\frac1{\ln^n x}$$