Solution for $a^x = b x^{-2}$

Question

I encountered an equation of the form $a^x = b x^{-2}$, where $a$, $b$ and $x$ are real numbers. But I am not sure if there is an analytical solution for this type of equations. I would appreciate it if you could point some directions as to where to look.

Answer 1

Assuming that $a$, $b$ and $x$ are real numbers, rewriting your equation as $$b=x^2a^x=(xe^{\frac{\ln{a}}{2}x})^2,$$ shows that there are no solutions if $b<0$. If $b>0$ then setting $w:=\tfrac{\ln{a}}{2}x$ yields $$we^w=\pm\tfrac{\ln{a}}{2}\sqrt{b},$$ the solutions of which are given precisely by the Lambert $W$ function; see Wikipedia for more information.