TITLE: Solutions of $x(\ln x)^2=e$
QUESTION [5 upvotes]: The solution of $x(\ln x)^2=e$ is clearly $e$, but how can I show that it is indeed a solution conceptually (that is, without substituting $x=e$) and that it is unique?

REPLY [2 votes]: $g(x)=x\log^2 x$ is a continuous and increasing function on $(1,+\infty)$, as a product of two continuous, non-negative and increasing functions on the same interval. Since $g(1)=0$ and $\lim_{x\to+\infty} g(x)=+\infty$, the map between $(1,+\infty)$ and $(0,+\infty)$ given by $g$ is one-to-one.