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Using calculus (or algebra), how would I solve an infinite exponent tower such as this?

$$c_0x^{c_1x^{c_2x^{c_3x^{.^{.^.}}}}}=a$$

Where $c_0=1$ and $c_{n+1}=\frac{c_n}{2}$ for $n=0,1,\ldots$ and $a>0$.

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    $\begingroup$ Mathjax is much nicer than HTML for math formatting. $\endgroup$ – suomynonA Apr 11 '17 at 4:56
  • $\begingroup$ Do I understand correctly that you want to know the result of the expression for $x=1$? $\endgroup$ – Fabian Apr 11 '17 at 5:13
  • $\begingroup$ As $1^\text{any thing}=1$, the answer is $a=1$. $\endgroup$ – Fabian Apr 11 '17 at 5:16
  • $\begingroup$ No, just a general formula to solve for 'x', with regards to a constant 'a' $\endgroup$ – Graviton Apr 11 '17 at 5:28
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    $\begingroup$ Note that $c_n = 2^{-n}$. $\endgroup$ – Patrick Stevens Apr 11 '17 at 6:44

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