Help solving non-trivial logarithmic inequality

Question

I have the following equation: $$\dfrac{2\pi G\lambda M^4}{m^2}\ln\left(\dfrac{\phi_i}{\phi_e}\right)+2\pi G\left(\phi_i^2-\phi_e^2\right)\ge 65$$

which, for the purpose of this question, I'll rewrite as: $$\ln{x}+ax^2\ge b$$

with $a$ and $b$ constants, and $x>0$.

I'd gladly appreciate any help solving this equation. I have thought about using the Lambert W function, but I can't get around the difference of powers in $x$. Any ideas?

---

EDIT: Following marty cohen's advice, I make the change of variable $y=x^2$ and get: $$\ln y+2ay\ge 2b\Longleftrightarrow 2ay\cdot e^{2ay}\ge 2a\cdot e^{2b}\Longleftrightarrow 2ay\ge W(2a\cdot e^{2b})$$

and finally: $$x\ge\sqrt{\dfrac{W(2a\cdot e^{2b})}{2a}}$$

Answer 1

Let $y = x^2$. The inequality becomes $\ln{y^{1/2}}+ay\ge b $ or $\ln y + 2ay \ge 2b $ so that Lambert can be used.