# Showing an inequality with ln

I have to show that the following inequation is true:

$\frac{\ln(x) + \ln(y)}{2} \leq \ln(\frac{x+y}{2})$

I transformed it into

$\frac{\ln(x \cdot y)}{2} \leq \ln(x+y) - \ln(2)$

because I thought that I better can show the inequation here, but I don't know how to proceed.

How can I proceed or am I completely wrong?

• You should try to have something like $\ln(X) \leq \ln(Y)$, and then use the fact that $\ln$ is an increasing function. – Esperluet Dec 5 '15 at 20:46
• Exponentiate both sides: $e^{\frac{\ln(x) + \ln(y)}{2}} \le e^{\ln(\frac{x+y}{2})}$ and see if you can proceed from here. – r9m Dec 5 '15 at 20:48

$$e^{(ln(x)+ln(y))/2}=e^{ln(x)/2}e^{ln(y)/2}=\sqrt{xy}$$
$$e^{ln((x+y)/2)}=(x+y)/2$$