For every $\displaystyle x \in (0,\infty) $ we have $\displaystyle ln'(x) = \frac{1}{x} $

So by the fundemental theorem of calculus we have $\displaystyle \int_{[a,b]}\frac{1}{x}dx=ln(b)-ln(a)$ for any interval $\displaystyle [a,b]$ in $\displaystyle (0,\infty)$

How can I proof this using the inverse function theorem?

Thanks