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15

Here's a question: if $x = \sqrt[3]{3}$, then what is the value of $y = x^{x^{x^{\dots}}}$? As you say, if $y$ is a solution to this, then $y \ln x = \ln y$, so that $$ \frac{\ln y}{y} = \ln(x) = \frac{\ln(3)}{3} $$ Now, how can we "solve this" for $y$? As it ends up, there are two solutions for $y$. The answer you are getting is the second root, $y ...


11

$$ x^{x^{x^{\kern3mu\raise1mu{.}\kern3mu\raise6mu{.}\kern3mu\raise12mu{.}}}}=y $$ $$ \ln{x^{x^{x^{\kern3mu\raise1mu{.}\kern3mu\raise6mu{.}\kern3mu\raise12mu{.}}}}}=\ln y $$ $$ x^{x^{x^{\kern3mu\raise1mu{.}\kern3mu\raise6mu{.}\kern3mu\raise12mu{.}}}}\ln{x}=\ln y $$ $$ y\ln{x}=\ln y $$ $$ \ln{x}=(\ln{y})/y $$ $$ \ln{x}=\ln({y^{1/y}}) $$ $$ x= y^{1/y} $$ ...



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