I am trying to understand the definition of a logarithm, because when I was trying to find the derivative of $2^x$ I got $2^x \lim_{h \to 0} \frac{2^h-1}{h}$ which I have found by searching to be $\ln(2)$. I did get a bit confused because I would need to use l'hopital rule, which would bring be back to what I was trying to find.
But my question that I think I need to understand before getting to my second question.
Euler defines logarithm as $\ln(x)=\lim_{n \to \infty}n(x^{\tfrac{1}{n}}-1)$
Which then must be equal to $\ln(x)=\lim_{h \to 0} \frac{x^h-1}{h} $
Could you help me understand how these are both the same?