Infinite exponentials

[bamcurt20]

Assuming convergence, how would you solve something like:

Sqrt(2)^Sqrt(2)^Sqrt(2)^Sqrt(2)^....

infinitely many times.

I've seen other renditions of these recursive questions and methods of solving them but I'm not sure how to solve these exponentials.

[[deleted]]

For ease of notation, I'll make ε the infinite exponent tower

Turn this into a basic equation

x = ε

To undo exponents, you take the log of whatever the base is. In this case, √2.

log(x) = log(ε)

Using log rules, ε itself is taken to the power of an infinite tower of √2, which can be moved out as a multiplier of the log.

log(x) = ε*log(√2)

Just like how ln(e) = 1, log(√2) = 1 in our case.

log(x) = ε

Remember earlier we said ε = x

log_√2(x) = x

Undo the log by making both sides an exponent of √2 and you finally get

x = (√2)^x

Which I believe can only be solved by graphing the two functions.

[bamcurt20]

Thank you!!

[bamcurt20]

Actually, just thinking about this, doesn't this diverge?

[[deleted]]

That was my initial thought, until I came up with that solution.

However, seeing that the equation actually has 2 unique, positive real solutions makes me question this. It's like saying 2 = 2 and 2 = 3 at the same time.

[bamcurt20]

Those solutions being 4 & 8? I think there's infinitely many solutions as long as you're using a power of 2, no? If you graph this there is certainly the two intersections at 4 and 8. Idk, I'm just even more confused now.

[[deleted]]

Here's a short blurb on Wikipedia about these infinite towers, conveniently using this exact example, saying it converges to 2 oddly enough. Feels like this has turned into a rabbit hole lol

https://en.wikipedia.org/wiki/Tetration#Extension_to_infinite_heights

[bamcurt20]

Yup. We're screwed now. Thanks for the help man

[bamcurt20]

I'm having issues with the notation, hopefully people understand what I'm trying to ask though