Lower bounds on the entropy function that go to infinity for $y \rightarrow 0$

Question

Let $D(x \| y)=x \ln \frac{x}{y}+(1-x) \ln \left(\frac{1-x}{1-y}\right)$ be the entropy function. In my case, I have $0.5 > x > y$. I need a reasonably tight lower bound with correct asymptotic behavior for $y \rightarrow 0$ (or as close as possible). Specifically, the lower bounds such as $\frac{3 (x-y)^2}{2(x+2y)}$ and others that paper on Wikipedia's page on Chernoff bounds all converge to a constant as $y \rightarrow 0$, whereas the correct behavior is that the function diverges to infinity. I am not certain what exactly I want the lower bound to look like, but I am looking for something that would be easier to work with, and logarithm-free. Are there any bound that could work for me? Any help would be much appreciated.