Of course, it is a well known fact that the inverse of $y=\ln x$ (natural logarithm of x) is $e^x$.
Assuming we haven't heard of the exponential function at all, how do we prove that the inverse of $\ln x$ i.e ($\ln^{-1} x$ ) is some other function, which indeed is the so called exponential function $e^x$?
Let me be a little more concrete. If $y = \ln x$, then $x= A^y$, how do I prove that $A= e$?
There's another method by which I tried to arrive at the inverse of natural logarithm of $x$.
By a theorem of differentiation of inverse functions
$\left.\dfrac{df}{dx}\right|_{x=a} \cdot\left.\dfrac{df^{-1}}{dx}\right|_{x=f(a)}=1$
I got an equation which looked like
$\left.\dfrac{dF(x)}{dx}\right|_{x=\ln k}=k$ where $k$ is a real number.
How do I actually show that $F(x)$ is that same exponential function $e^x$?
Is there any way of solving such an equation, or such a problem? Or am I just talking nonsense? I would love to be enlightened by all of you.
Thanks in advance. :)