Thank you for the answers to my earlier question. By working through all your answers, I arrive at a more refined question below.
Fix a positive real constant $b$. Let $x_0$ be the smallest nonnegative real number such that the inequality $2^x\ge bx$ holds for all $x\ge x_0$. Does there exist a constant $C>0$ such that $x_0≤C \log_2 b$?
I showed that $x_0$ has order $O(b^\epsilon)$ in my workings in the earlier question.