# What is x^x^x^x...

• Jul 31st 2012, 12:34 AM
jgv115
What is x^x^x^x...
How would I find the solution, or simplify $x^{x^{x^{x...}}}$
• Jul 31st 2012, 02:28 AM
JJacquelin
Re: 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 PM
Soroban
Re: What is x^x^x^x...
Hello, jgv115!

Quote:

How would I find the solution, or simplify $x^{x^{x^{x...}}}$
You can't . . .

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

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

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

And we have: . $\ln(y) -y\ln(x) \:=\:0$
. . a transcendental equation in $y.$
We cannot solve for $y.$

By the way, JJ is incorrect.

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

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

To change the order, parentheses are required: . $(4^3)^2 \:=\:64^2 \:=\:4,\!096$
• Jul 31st 2012, 10:01 PM
JJacquelin
Re: What is x^x^x^x...
Quote:

Originally Posted by Soroban
Hello, jgv115!

You can't . . .

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

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

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

And we have: . $\ln(y) -y\ln(x) \:=\:0$
. . a transcendental equation in $y.$
We cannot solve for $y.$

By the way, JJ is incorrect.

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

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

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

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&#40;x&#41; - Wolfram|Alpha