# How can I solve this infinite exponent tower?

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 $a>0$.

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