Halley’s formula

The following formula is due to the English scientist and mathematician Edmond Halley (1656 à 1742):

$\ln x=\underset{n\to \mathrm{\infty }}{lim}(\sqrt[n]{x}-1)n$(1)

Proof.  We change the $n$th root to power of $e$ and use the power series expansion of exponential function:

$(\sqrt[n]{x}-1)n$$\mathrm{\hspace{0.33em}}=(e^{\frac{\ln x}{n}}-1)n$

$\mathrm{\hspace{0.33em}}=\left({\displaystyle \sum _{m=0}^{\mathrm{\infty }}}{\displaystyle \frac{{(\ln x/n)}^m)}{m!}}-1\right)n$

$\mathrm{\hspace{0.33em}}={\displaystyle \sum _{m=1}^{\mathrm{\infty }}}{\displaystyle \frac{{(\ln x/n)}^{m}n}{m!}}$

$\mathrm{\hspace{0.33em}}=\ln x+{\displaystyle \frac{1}{n}}{\displaystyle \sum _{m=2}^{\mathrm{\infty }}}{\displaystyle \frac{{(\ln x)}^m}{m!n^{m-2}}}$

The last converging series has a finite sum, and as  $n\to \mathrm{\infty }$,  the asserted formula follows.

Note.  The formula (1) was known also by Leonhard Euler, who used it for defining the natural logarithm.  Using (1), one can easily prove the well-known laws of logarithm, e.g.

$\ln xy$$\mathrm{\hspace{0.33em}}=\underset{n\to \mathrm{\infty }}{lim}(\sqrt[n]{x}\sqrt[n]{y}-1)n$

$\mathrm{\hspace{0.33em}}=\underset{n\to \mathrm{\infty }}{lim}(\sqrt[n]{x}\sqrt[n]{y}-\sqrt[n]{y}+\sqrt[n]{y}-1)n$

$\mathrm{\hspace{0.33em}}=\underset{n\to \mathrm{\infty }}{lim}{y}^{\frac{1}{n}}(\sqrt[n]{x}-1)n+\underset{n\to \mathrm{\infty }}{lim}(\sqrt[n]{y}-1)n$

$\mathrm{\hspace{0.33em}}=y^0\ln x+\ln y$

$\mathrm{\hspace{0.33em}}=\ln x+\ln y.$