Why does $\oint_C d\log|f(z)|=0$?

Question

In the article on the argument principle on planetmath , it says that $$\oint_C d\log|f(z)|=0$$ since $|f(z)|$ is single-valued. Why does that follow, or can someone point me to a fuller explanation? I'm studying complex-analysis right now, but this result is not obvious to me. Thanks.

Answer 1

Since $\log|f(z)|$ is single-valued, we have $$ \oint_Cd\log|f(z)|=\log|f(z)|\bigg|_A^A=0 $$ where $A$ is any point on $C$.