Logarithmic Inequalities.

Quote:

Originally Posted by Kaloda

For which real numbers x does inequality $2\log_x\frac{a+b}{2}\leq\log_x(a)+\log_x(b)$
holds for all positive numbers a and b?

Here are some observations. For $\log_x(t)$ to be defined it is necessary for $t>0$ and $0<x<1\text{ or }x>1$ .

The function $\log_x(t)$ is one-to-one and it is increasing if $x>1$ and decreasing for $0<x<1$ .

So this is a concept question.
Compare $\left(\frac{a+b}{2}\right)^2\le\text{ or }\ge a\cdot b$ .