<s>[INST] Prove inequality with logarithms:$\log_{a}{\frac{a+b}{2}} + \log_{b}{\frac{a+b}{2}} \ge 2$ Let $ a, b \in (1, \infty)$. Prove that:
$$\log_{a}{(\frac{a+b}{2})} + \log_{b}{(\frac{a+b}{2})} \ge 2$$
I tried switching the bases on each of the logaritm but I got stuck:
$$\frac{\log_{\frac{a+b}{2}}{(ab)}}{\log_{\frac{a+b}{2}}{(a)}\log_{\frac{a+b}{2}}{(b)}}$$
 [/INST] By AM-GM we have $$\frac{a+b}2\geq \sqrt{ab} $$ and since $\log$ is an increasing function for bases $>1$ we have $$\log_a(\frac{a+b}2) +\log_b(\frac{a+b}2) \geq \log_a(\sqrt{ab}) +\log_b(\sqrt{ab}) =1/2(2+\log_a(b)+\log_b(a))$$
And we have that $\log_a(b) +\log_b(a) =\log_a(b) +\frac1{\log_a(b)} \geq 2$ because $x+\frac1 x \geq 2$
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