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)?

x-1 is a crude approximation that is good for values of x close to 1.
2*(x-1)/(x+1) is a better approximation with a larger range.
6*(x-1)/(1 + 4 * Sqrt(x) + x) is better and has an even larger range.
90*(x-1)/(7+32 * x^(1/4) + 12 * x^(1/2) + 32 * x^(3/4) + 7 * x) is even better and has a larger range, but it never exceeds 90 for any positive real value of x.
Some such approximations are better than others, but any of these fail for sufficiently large x.
(Note that the fourth root in that last formula can be found by taking the square root twice).
