ln(x+y)=ln(x)+ln(y)

Question

The question demands an answer to when ln(x+y) can be considered to equal ln(x)+ ln(y) and we are asked to use contour to prove this but the concept of contour does not entirely make sense to me. Anyways the answer is for y=x/(x-1) for x>1 how do we reach this using contour?

Answer 1

HINT: Observe $$\begin{align}\ln(x + y) &= \ln(x) + \ln(y)\\ &= \ln(xy) \tag{Product Log Rule}\\ \implies x + y &= xy \tag{One-to-one property} \end{align}$$

Can you figure out what to do from here?

Answer 2

$$\ln(x+y) = \ln x+\ln y$$ Use the Product Rule.
$$\implies \ln(x+y) = \ln(x\cdot y)$$ Remove the natural log from both sides.
$$\implies x+y = xy$$ $$x = xy-y \implies x = y(x-1) \implies y = \frac{x}{x-1};x>1$$ Very clearly, $x$ cannot be $0$ (incorrect) or $1$ ($y$ becomes undefined), so our domain is $x>1$.