# Math Help - What is x^x^x^x...

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

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

2. ## 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
Attached Thumbnails

3. ## Re: What is x^x^x^x...

Hello, jgv115!

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: . $(4^3)^2 \:=\:64^2 \:=\:4,\!096$

4. ## Re: What is x^x^x^x...

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
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 20, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.949283242225647, "perplexity": 2861.733512820868}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-48/segments/1448398464253.80/warc/CC-MAIN-20151124205424-00138-ip-10-71-132-137.ec2.internal.warc.gz"}