TITLE: Inequality $\log x\le \frac{2}{e} \, \sqrt{x}$
QUESTION [6 upvotes]: The inequality $$\log x \le \frac{2}{e} \, \sqrt{x},$$ where $\log x$ denotes the natural logarithm, is used in the proof of Theorem 4.7 in Apostol's Analytic Number Theory.
It seems that the inequality is not very difficult to prove using calculus. We could simply find maximum/minimum of some function like $f(x)= \frac2e \sqrt{x} - \log x$ or $g(x)=\frac{\log x}{\sqrt x}$.
Are there some other methods how this inequality can be proved? Is there a way in which this inequality can be seen more directly, without having to calculate critical points of some auxiliary function?

REPLY [3 votes]: Letting $x=(eu)^2$, the inequality is equivalent to
$$\log u\le u-1$$