What is the method to correctly isolate $y$ as the dependent variable for $x = e^y$?

Question

In this youtube video about 5:00 minutes in, the instructor makes the point that you can simply exchange the $x$ and $y$ values of the exponential form $x = e^y$ of the equation $y = ln x$ to make $y$ the dependent variable. In other words, he was saying that you can just arbitrarily make the following re-arrangement.

$x = e^y \implies y = e^x$

I would really like to understand specifically how this re-arrangement is possible.

For example, if I let $x = 2$. then $ln 2 = 0.6931$, re-arranging to exponential $e^.6931 = 2$ however $e^2 \ne 0.6931$, thus my confusion on this point.

My question is, how do I get from ($y = ln x \implies x = e^y$) to $y = e^x$?

Answer 1

$x=e^y$ does not imply $y=e^x$, because if it did, then for any finite number $x\in \mathbb{R}$,

$$x=e^y = e^{e^x}=e^{e^{e^y}} =e^{e^{e^{e^{x}}}} = \cdots = \infty $$

Which is a contradiction (i.e. non-sense).

What you really want is,

$$x=\ln (y)$$ implies $$e^x = e^{ln(y)}$$

And then remember that exponential and natural log ''cancel'' i.e. ($e^{\ln(z)}=z=\ln(e^z)$)

So

$$e^x=y$$

Answer 2

Considering @cooper's comment and your first comment above:

The function $y=\exp(x)$ and the relation $x=\exp(y)$ are defined differently. Just look at their plots:

But sometimes we change the alphabet $x$ with $y$ just to read the relation we got easily. For example, when we want to find the inverse of an strictly increasing function $y=f(x)$ we do some manipulation to get a relation like $x=g(y)$ and then we change the alphabets $x$ and $y$. So we'll write $y=g(x)$ knowing that this new $y$ is in fact $y^{-1}$ of the first $y$. What we faced at this video, I think, is very similar to what I noted above.