How to sketch the level curves of the following function

Question

Consider the function f defined by $f(x,y)=ln(x-y)$

How do I sketch the level curves for this function for the values of $k=-2,0,2$?

Answer 1

Observe that the level curves of $f(x, y)$ are given by $f(x, y) = k$ for some constant $k.$ Considering that $f(x, y) = \ln(x - y),$ the level curves of $f(x, y)$ are given by $\ln(x - y) = k.$ Of course, this is equivalent to $e^k = x - y,$ and this is equivalent to $y = x - e^k.$ Consequently, the level curves are the family of parallel lines $y = x - e^k$ with slope $1$ and $y-$intercept $e^k$ for each value of $k.$