So I was looking at the function $y = \ln(x+\ln(x+\ln(x+\ln(x...$
I was thinking, how could someone possibly evaluate this??
Then, I noticed that if I removed a log, I would have $e^y = x+y$.
Boom, easy closed expression.
Now, I think, under what circumstances can I take a CLOSED FORM relationship and convert it to a format like above?
Like for example, I can write $y = \sqrt{2-x\sqrt{2-x\sqrt{2-x\dots}}} $ as $y^2=2-xy$, but what tells me that I can write the latter as the first??
What lets you do this is merely the fact that:
If you take something from infinity ($\infty)$, then what you have left is still infinity.
This is the main concept that is involved in this type of calculations.
As for this equation of yours, $$y = \sqrt{2-x\sqrt{2-x\sqrt{2-x\dots}}} $$ $y$ is an infinite series of radicals.
First you square both sides, then you subtract $2$ from both sides and then you divide both sides by $-x$. Even after all these operations, what you've got is still an infinite series of radicals which as you have defined above is $y$.