Solving Some Transcendental Equations

Question

How do you solve for $a$ in each of the equations $$a^{a^a}=b^c$$ $$a^{a^{a^a}}=b^c$$ $$a^{a^{a^a}}=b^{c^d}?$$

Answer 1

Since b, c, d are independent of a, the right hand term can be replaced with a single symbol which represents a constant. Let this symbol be A. We have $a^{a^a}=A\iff a^a\ln a=\ln A\iff a\ln a$ $+\ln\ln a=\ln\ln A$. By letting $a=e^t$, we have $t\cdot e^t+\ln t=\ln\ln A$, which obviously isn't even expressible in terms of the Lambert W function, since it can only solve equations of the form $t\cdot e^t$ = constant, but, in this case, $\ln\ln A-\ln t\neq$ constant, since it depends on t. Which of course does not mean that names or notations for a do not exist, or that its value cannot be found by numerical methods: they are called super-roots or super-radicals, but they possess an analytical solution only for the case $n=1$ or $n=2,$ where n represents the height of the power-tower of a's. This operation of repeated exponentiation is called tetration, by the way.