To expand the function $f(x, y) = e^{-x} \log(y)$ up to the second degree, we need to use a Taylor series expansion around a point, typically $(x_0, y_0)$. Let’s assume we expand around the point $(x_0 = 0, y_0 = 1)$, which is a common choice for logarithmic functions.

The general form of the second-degree Taylor expansion for a function of two variables $f(x, y)$ around $(x_0, y_0)$ is:

$f(x, y) \approx f(x_0, y_0) + (x - x_0) \frac{\partial f}{\partial x} \bigg|_{(x_0, y_0)} + (y - y_0) \frac{\partial f}{\partial y} \bigg|_{(x_0, y_0)} + \frac{1}{2} \left( (x - x_0)^2 \frac{\partial^2 f}{\partial x^2} \bigg|_{(x_0, y_0)} + 2(x - x_0)(y - y_0) \frac{\partial^2 f}{\partial x \partial y} \bigg|_{(x_0, y_0)} + (y - y_0)^2 \frac{\partial^2 f}{\partial y^2} \bigg|_{(x_0, y_0)} \right)$

Step 1: Calculate $f(x_0, y_0)$

$f(0, 1) = e^0 \log(1) = 1 \cdot 0 = 0$