Find $x$ such that $(ax)^{bx}>c$, where $a,b,c,x>0$

Question

Let $a>0$, $b>0$, and $c>0$. Let $$x_0\triangleq \inf\{x>0:{\rm for~all~} \bar x>x, (a\bar x)^{b\bar x}>c\}.$$ What is a good estimate (least-conservative estimate) for $x_0?$

Answer 1

You can begin by simplifying the inequality like this: $$(a x)^{b x}>c$$ set $a x=y$ so $x = y/a$ $$y^{by/a}>c$$ now raise to the power $a/b$ to get $$y^y>c^{a/b}$$ Calling $d = c^{a/b}$ your inequality simplifies to $$y^y>d$$

As already was observed, this leads to solve the equation $y^y=d$, whose solution is given by $y = \frac{\log(d)}{W(\log(d))}$ where $W$ is the Lambert function.

Can you take it from here?