1. **State the problem:** Determine the degree of homogeneity of the function $$f(x,y) = x \ln(x) - y \ln(y)$$. 2. **Recall the definition:** A function $$f(x,y)$$ is homogeneous of degree $$k$$ if for all $$t > 0$$, $$$f(tx, ty) = t^k f(x,y)$$$. 3. **Apply the definition:** Calculate $$f(tx, ty)$$: $$$f(tx, ty) = (tx) \ln(tx) - (ty) \ln(ty) = t x (\ln t + \ln x) - t y (\ln t + \ln y)$$$ 4. **Distribute terms:** $$$= t x \ln t + t x \ln x - t y \ln t - t y \ln y = t \ln t (x - y) + t (x \ln x - y \ln y)$$$ 5. **Compare with $$t^k f(x,y)$$:** The original function is $$f(x,y) = x \ln x - y \ln y$$. If $$f$$ were homogeneous of degree $$k$$, then $$$f(tx, ty) = t^k f(x,y) = t^k (x \ln x - y \ln y)$$$ 6. **Check if the expression matches:** From step 4, $$$f(tx, ty) = t \ln t (x - y) + t f(x,y)$$$ This is not equal to $$t^k f(x,y)$$ for any constant $$k$$ because of the extra term $$t \ln t (x - y)$$. 7. **Conclusion:** The function is **not homogeneous**. **Final answer:** B Not Homogeneous