Question

Which is smaller? $2$ or ${\log }_{\pi }2+{\log }_2\pi$

Answer 1

We know, $AM\ge GM$

$\frac{{\log }_{\pi }2+{\log }_2\pi }{2}\ge \sqrt{{\log }_{\pi }2\times {\log }_2\pi }$

$\left({\log }_{\pi }2\right)+{\log }_2\pi \ge 2$

hence, $2$ is less than ${\log }_{{\pi }^2}+{\log }_2\pi$