Yet Another Power Tower

Calculus Level 4

\[\huge a^{a^{a^{\cdot^{\cdot^\cdot}}}}=\frac{1}{3}\]

How many positive real solutions \(a\) does the above equation have?

Hint: Argue "from first principles", without using any properties of the infinite power tower you might know.

Clarification: The value of the infinite power tower \(a^{a^{a^{.^{.^.}}}}\) is defined as the limit of the sequence \(x_1=a, x_{n+1}=a^{x_n},\) if that limit exists.


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