counter example for $\mathbb{E}[\log(x)] <\mathbb{E}[\log(y)]$ when $\mathbb{E}[x]< \mathbb{E}[y]$ and vice versa

Question

I am looking for an example that $\mathbb{E}[x]< \mathbb{E}[y]$ but not $\mathbb{E}[\log(x)] <\mathbb{E}[\log(y)]$ and an example that $\mathbb{E}[\log(x)] <\mathbb{E}[\log(y)]$ but not $\mathbb{E}[x]< \mathbb{E}[y]$.

Is there any such example? Or are they equivalent conditions?

Please note we can assume both $x,y\geq 0$.

Answer 1

Suppose $x=1$ with certainty and $y$ is equal to $0.4$ or $2$ with equal probability. Then $$ \mathbb E[x] = 1 \lt 1.2 = \mathbb E[y],$$ and $$ \mathbb E[\log(x)] = 0 \gt -0.11157 \approx \mathbb E[\log(y)].$$ Flip $x$ and $y$ and you get your other example.