Verification of a logarithmic inequality

Question

Verify the inequality $ \frac{(\log (x) + \log (y))}{2} \le \log\frac{(x+y)}{2}$, where $x,y>0$

I'm still struggling how to solve the inequality, I have tried AM-GM and Bernoulli, without any success.My suggestion is that the solution is very elementare, but I can't see it.

Answer 1

$\log$ is a concave function, which says that your inequality is true by Jensen.

About Jensen see here: https://en.wikipedia.org/wiki/Jensen%27s_inequality

Answer 2

You have\begin{align}\frac{\log(x)+\log(y)}2&=\frac12\log(xy)\\&=\log\left(\sqrt{xy}\right)\\&\leqslant\log\left({\frac{x+y}2}\right).\end{align}