Approximating logarithms of large numbers

I recently needed to approximate a logarithm of the form $\ln(x)$ where $x$ is some large number. It was not possible to use the usual Maclaurin series approximation for $\ln(1+x)$ because this only holds for $-1 < x \leq 1$ . However, the following is a useful trick. We have

$\ln(x) = \int_1^x \frac{d y}{y}$

Therefore, suppose we replace $\frac{1}{y}$ with $\frac{1}{y^{1-\frac{1}{n}}}$ , where $n \gg 1$ is any large number. Then

$\ln(x) \approx \int_1^x \frac{dy}{y^{1-\frac{1}{n}}}$

$= \int_1^x dy \big(y^{\frac{1}{n}-1}\big) = \bigg[\frac{1}{(1/n)}y^{1/n}\bigg]_1^x$

$= nx^{1/n}-n$

The approximation formula $\ln(x) \approx nx^{1/n}-n$ works surprisingly well, being more accurate the larger is $n$ . For example, to calculator-accuracy we have $\ln(10^6) = 13.81551056$ , while taking $n=100000$ we get the approximation $100000 \times (10^6)^{1/100000}-100000 = 13.81646494$ .