It is not difficult to show that $\ln(x)$ for real positive $x$ can be approximated by the series 
$$
-\gamma+\sum\_{k=1}^\infty \frac{(-1)^{k+1}}{k!k}x^k
$$
up to an absolute error bounded by $e^{-x}/x,$ where $\gamma$ is the Euler-Mascheroni constant. I assume this must be well known and would appreciate any references. 



Approximating logarithm by a series for large values
The statement is equivalent to an expansion for the [Exponential Integral function](https://en.wikipedia.org/wiki/Exponential_integral#Convergent_series) $\operatorname{Ei}(x)$, so any text on special functions should do (check e.g. the references in the linked wiki article).


