Thread: What is x^x^x^x...

How would I find the solution, or simplify

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"
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Hello, jgv115!

How would I find the solution, or simplify

You can't . . .


Let

Then: .

Take logs: .

And we have: .
. . a transcendental equation in
We cannot solve for


By the way, JJ is incorrect.

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


For example, means: .

To change the order, parentheses are required: .

Originally Posted by Soroban

Hello, jgv115!

You can't . . .


Let

Then: .

Take logs: .

And we have: .
. . a transcendental equation in
We cannot solve for


By the way, JJ is incorrect.

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


For example, means: .

To change the order, parentheses are required: .

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