Detailed proof on the limit expression of natural logarithm

Question

I want to know the derivation of the following identity:

$$ \ln({x}) = \lim_{n \to \infty} n(x^{1/n} - 1) $$

I've checked this post for answers but I'm having a hard time grasping the explanations. Especially

"$\ln({z})$ is the derivative of $t\mapsto z^t$ at $t=0$"

Can anyone explain it clearly to me? Thanks.

Answer 1

$$\lim_{n\to \infty }n(x^{\frac{1}{n}}-1)=\lim_{n\to \infty }\frac{e^{\frac{1}{n}\ln(x)}-1}{\frac{1}{n}}=\left.\frac{\mathrm d }{\mathrm d y}\right|_{y=0}e^{y\ln(x)}=\ln(x).$$