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$ x^{x^{x^{x}}}=5 $

Here is what tetration is about

Note : numerically method isn't what I want. (Sorry for I didn't emphasize it before.)

giao
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    Could you clarify what you mean by "analytically"? From what set is $x$ allowed to take values from? The Reals? – Albert Dec 30 '20 at 14:59
  • The Lambert-W-function might "save the day" , if not you will have to apply numerical methods (the Lambert-W-function , strictly speaking, is also a numerical method). Even $x^x=5$ cannot be solved "directly" (for example like a quadratic equation). And this equation is vastly more complicated. – Peter Dec 30 '20 at 15:06

2 Answers2

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$$f(x)=x^{x^{x^x}}-5$$ we find a zero with Newton's method $$f'(x)=x^{x^{x^x}} \left(x^{x^x-1}+x^{x^x} \log x \left(x^{x-1}+x^x \log x (\log x+1)\right)\right)$$ We set $x_0=1.5$ and $x_{n+1}=x_n-\frac{f(x)}{f'(x)}$

I got the following results

$$ \begin{array}{r|r|r} n & x_n & error \\ \hline 0 & 1.5 & \\ 1 & 1.83748 & 0.337477 \\ 2 & 1.79603 & 0.0414445 \\ 3 & 1.74805 & 0.0479864 \\ 4 & 1.70007 & 0.0479734 \\ 5 & 1.66681 & 0.0332666 \\ 6 & 1.65618 & 0.0106251 \\ 7 & 1.65539 & 0.000786085 \\ 8 & 1.6553947522 & 3.85\times 10^{-6} \\ 9 & 1.6553909024 & 9.16\times 10^{-11} \\ \end{array} $$

Thus the solution with $9$ exact decimals is $x\approx 1.655390902$.

Raffaele
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If $$x=\sin \left(\frac{8 \pi }{35}\right)+\cos \left(\frac{\pi }{44}\right)$$ $$x^{x^{x^x}} =4.9999978$$

Claude Leibovici
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