How would I find the solution, or simplify $\displaystyle x^{x^{x^{x...}}}$

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- Jul 31st 2012, 12:34 AMjgv115What is x^x^x^x...
How would I find the solution, or simplify $\displaystyle x^{x^{x^{x...}}}$

- Jul 31st 2012, 02:28 AMJJacquelinRe: What is x^x^x^x...
Hi !

it is an "infinite power tower"

But your symbolism is ambiguous : it could means several different power towers. One have to use brakets to give a precise definition (in attachment)

If the tower is finite (with a finit number of x), the related function of x can be formally expressed as the first derivative of a Generalized Sophomores Dream function : Formula 12:5 in the paper "The Sophomores Dream Function"

Scribd - Jul 31st 2012, 02:39 PMSorobanRe: What is x^x^x^x...
Hello, jgv115!

Quote:

How would I find the solution, or simplify $\displaystyle x^{x^{x^{x...}}}$

Let $\displaystyle y \;=\;x^{x^{x\hdots}} $

Then: .$\displaystyle y \:=\:x^y$

Take logs: .$\displaystyle \ln(y) \:=\:\ln\left(x^y\right) \quad\Rightarrow\quad \ln(y) \:=\:y\ln(x)$

And we have: .$\displaystyle \ln(y) -y\ln(x) \:=\:0$

. . a*transcendental*equation in $\displaystyle y.$

We can__not__solve for $\displaystyle y.$

By the way, JJ is incorrect.

An exponential stack is always read "from the top down".

For example, $\displaystyle 4^{3^2}$ means: .$\displaystyle 4^9 \:=\:262,\!144$

To change the order, parentheses are required: .$\displaystyle (4^3)^2 \:=\:64^2 \:=\:4,\!096$

- Jul 31st 2012, 10:01 PMJJacquelinRe: What is x^x^x^x...
Hi Soroban !

you are right about the conventional hierarchy of operations.

By the way, formal solving (for y) of the equation ln(y)-y*ln(x) is :

y = -W(-ln(x))/ln(x) where W is the Lambert function.

As others special functions, the LambertW function cannot be expressed as combination of a finite number of elementary functions.

W(x) - Wolfram|Alpha