For which real numbers x does inequality $\displaystyle 2\log_x\frac{a+b}{2}\leq\log_x(a)+\log_x(b)$
holds for all positive numbers a and b?
Here are some observations. For $\displaystyle \log_x(t)$ to be defined it is necessary for $\displaystyle t>0$ and $\displaystyle 0<x<1\text{ or }x>1$.
The function $\displaystyle \log_x(t)$ is one-to-one and it is increasing if $\displaystyle x>1$ and decreasing for $\displaystyle 0<x<1$.
So this is a concept question.
Compare $\displaystyle \left(\frac{a+b}{2}\right)^2\le\text{ or }\ge a\cdot b$.