# Tag Info

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