using $\epsilon-\delta$ to prove the continuity of a multivariable function

Question

How can I prove $f(x,y)=x^y$ is continuous on $U=\{(x,y)\in\mathbb{R}^2|x>0\}$ by using $\epsilon-\delta$?
For $(x_0,y_0)\in U$, I tried to do something with $y\ln x$: $$y\ln x-y_0\ln x_0=(y-y_0)\ln x+y_0\ln\frac{x}{x_0}.$$ I want $x-x_0$ to occur in the equality, so that I can choose $\delta$ from $\sqrt{(x-x_0)^2+(y-y_0)^2}<\delta$. But I don't know how to do it :( . Could someone help me?