Say we have $y=e^x$, we then apply to both sides $\ln$ and hence get: $\ln y=x$....apparently from there we get to $y=\ln x$ somehow..although I cannot think of any logarithm law which should yield this..?
The aim of this operation was to find the inverse function of $y=e^x$.
We start with $y=e^x$. Taking the $\ln$ of both sides gives us $\ln y=\ln e^x$.
Using the logarithm property that $\ln a^b=b\ln a$, we simplify to:
$\ln y=\ln e^x\implies \ln y=x(\ln e)\implies \ln y=x(1)=x$
It should not be $y=\ln x$
We can see that $\ln y=x$ does not mean that $y=\ln x$ in general with a counterexample. Consider $y=e$, and $x=1$. In the first equation those values work, but in the second equation $e\neq\ln1=0$
To find the inverse of $e^x$:
We start with $y=e^x$. We switch the $x$ and the $y$, giving us $x=e^y$. Now we solve for $y$. Taking the $\ln$ of both sides gives us, $\ln x=y\ln e=y$.
Hence the inverse of $f(x)=e^x$ is $f^{-1}(x)=\ln x$