Is there an approximation to the natural log function at large values? 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)?

The taylor series for $\ln(\cdot)$ in the vicinity of point $a$ is
$$\ln(x+a) =\ln(a) + 2 \cdot (y+\frac{y^3}{3}+\frac{y^5}{5}+\frac{y^7}{7} + \ldots + \frac{y^{2n+1}}{2n+1}), \mathrm{where\;} y=\frac{x}{x+2a}$$
So for fast convergence, we need the logarithms of first 26 primes numbers: $2,3,5,7.\ldots$
