# Tower of exponents solution approach unique to exponents?

1. Feb 4, 2016

### DocZaius

I saw a YouTube video presenting what seemed like a clever solution to $x^{x^{x^{.^{.}}}} = 2$ (which is to say: an infinite tower of exponents of x = 2). He said to consider just the exponents and ignore the base and realize that those exponents themselves become a restatement of the whole left side of the equation. Equate them to 2 and you get $x^2=2$ and thus $x= \sqrt{2}$.

I thought that was clever and tried to see if this could work for other operations like multiplication ($x\cdot x\cdot x \cdot\ ...=2$), division ($\frac{x}{\frac{x}{\frac{x}{...}}}=2$), addition ($x+x+x+...=2$) and subtraction($x-x-x-...=2$), but realized it didn't. Any idea what's special about the exponent operation that allows for this clever "substitution of infinite terms" type of solution?

2. Feb 4, 2016

### Staff: Mentor

The exponent tower can have a limit, the other processes do not (apart from boring special cases).

That should work as well:
$$x\sqrt{x\sqrt{x\sqrt \dots}}$$

3. Feb 5, 2016

### Samy_A

Even for the infinite tower of exponents the trick will only work if $e^{-e}<x<e^{1/e}$. As $e^{1/e}=1.444...$ and $\sqrt 2=1.414...$, it works for $\sqrt 2$.
https://thatsmaths.files.wordpress.com/2013/01/powertowerlambert.pdf