# Logarithmic Inequalities.

• February 13th 2013, 04:06 AM
Kaloda
Logarithmic Inequalities.
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?
• February 13th 2013, 06:39 AM
Plato
Re: 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 $01$.

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

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