Two definitions of the natural logarithm

Question

Apart from the definition involving integration, I would like to check whether the following definitions of the $\ln$ function are :

1. accurate
2. equivalent

Def 1 .$$\begin{cases} (\forall x, y \in (0,+\infty)) \quad f(xy) = f(x)+f(y) \\ f \text{ is differentiable at 1 with } f'(1)=1 \end{cases}$$

Def 2 $$ \begin{cases} f \text{ is differentiable on } (0,+\infty) \text{ with } f'(x) = \dfrac 1x \text{ for all } x \in (0,+\infty) \\ f(1)=0\end{cases}$$

Answer 1

In both cases, my answer is: it is fine, as long as you prove that such a function exists. Assuming that, yes, these definitions are equivalent to $\log(x)=\int_1^x\frac{\mathrm dt}t$.