TITLE: Is there an approximation to the natural log function at large values?
QUESTION [18 upvotes]: At small values close to $x=1$, you can use taylor expansion for $\ln x$:
$$ \ln x = (x-1) - \frac{1}{2}(x-1)^2 + ....$$
Is there any valid expansion or approximation for large values (or at infinity)?

REPLY [10 votes]: for any positive $x$
$\ln \left( x \right) \approx a{x^{\frac{1}{a}}} - a$
where $a$ is any large constant 
The larger $a$, the better approximation.