Implicit equation $\ln(\frac{x}{y})-y=1$ to rectangular equation not in terms of $W(x)$

Question

Backstory and Other Info

I'm not sure if this is possible, I'm currently a precalculus student and have a very limited understanding of much of any of this. However, I do like to go on WolframAlpha and do all out random things. One of the things I tried to do was to take one of the functions that approximates e, or Euler's number:

$$y=(1+\frac{1}{x})^{x}$$

which has the property

$$\lim_{x \to \infty}{(1+\frac{1}{x})^{x}=e}$$

and find the inverse, $f^{-1}(x)$, of it. When I asked WolframAlpha for the inverse, it gave me this equation:

$$ -\frac{\ln{x}}{W(\frac{-\ln{x}}{x}) + \ln{x}} $$

This is how I discovered the product log.

A quick search on Wikipedia showed me this definition:

$W(xe^{x})=x$

I also noted that there are multiple branches. Intuitively, I think I understand what a branch is, but I don't know the exact definition. Some help conceptualizing this is helpful, but not necessary.

The Question

From the definition given about $W(x)$, I derived the implicit equation $\ln(\frac{x}{y})-y=1$ using my basic algebra and precalculus knowledge.

From this implicit equation, I wanted to derive a rectangular function, $y=f(x)$, of the branch

$W_0(x)\space$ for $\space x\ge-1$

All points of the above pass the vertical line test.

Still, not entirely sure if this is possible, and I may be ignorant for asking. I would prefer it to be a function only using algebraic concepts, because my precalculus class hasn't actually taught us much calculus, just the algebra we would need for calculus, limits, and limits to infinity.

Answer 1

Here is how to convert back to $W$ notation.

$$\ln \dfrac{x}{y}-y = 1$$

$$\ln \dfrac{x}{y} = 1+y$$

$$\dfrac{x}{y} = e^{1+y}$$

$$x = ye^{1+y} = eye^y$$

$$\dfrac{x}{e} = ye^y$$

$$y = W\left( \dfrac{x}{e}\right)$$

Note: $W(x)$ is just a function of $x$. So, you asking if it is possible to represent a solution without using $W$, it is not entirely clear what you mean. Do you want a power series representation? Based on your comments, that appears to be what you are looking for, so here is the answer using a power series:

$$y = \sum_{n\ge 1} \dfrac{(-1)^{n+1}n^{n-2}x^n}{(n-1)!e^n}$$