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?
How about we exponentiate both sides? We get
$$e^{(ln(x)+ln(y))/2}=e^{ln(x)/2}e^{ln(y)/2}=\sqrt{xy}$$
and
$$e^{ln((x+y)/2)}=(x+y)/2$$
Now the result is immediate per the AM-GM inequality.