Convexity of this function, the ordinary definition of convex functions took me a lot of time!

Question

I have been struggling with this problem for about 2 days, I could prove that following function is convex using the definition of convex functions, but I need to prove its convexity using Bregman Distance. so far I know that it is a Bregman Distance of another function, but I could not find that function. I appreciate any help, thanks.
$$ \left\{ \begin{array}{cc} (x_1 -x_2) (\ln(x_1) -\ln(x_2))\, \,\,\,\mbox{ if} \,\,\,x \in R^2_{++} \\ 0 \qquad \qquad \mbox{ if},\, x=0 \\ +\infty \qquad \qquad \mbox{ otherwise} \end{array} \right. $$

Answer 1

Hint : what is the Bregman distance of $x\ln x$ (linear functions are convex)?