Showing an inequality with ln

I have to show that the following inequation is true:

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

I transformed it into

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

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?

I have to show that the following inequation is true:

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

I transformed it into

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

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?