Define

$\mathrm{ln}(x)={\int}_{1}^{x}\frac{1}{t}$

Assume I have proven that $\mathrm{ln}x$ is one-to-one and therefore has an inverse $\mathrm{exp}(x)$

Define $e$ as:

$\mathrm{ln}e=1$

Now, if you have no other notion of exponentials, or logarithms, how could define what ${e}^{x}$ means and show that its the inverse of $\mathrm{ln}x$?

You are allowed to assume the logarithmic product and quotient property.

Thanks for the help.