Analiytic Continuation of a function

Question

For the given curve $γ$ fix the regular in domain C\ $γ$ branch of $f(z) = ln z$ by $f(1) = 0$. For that branch find the value of its analytic continuation f(−3) =?

I just know that we have to work on the argument. I already have : $ln(z) = ln|z| + iarg(z)$. Since f(1) = 0, arg(0) = 0. But as with this curve we never touch (-3)

Answer 1

Within the region we can travel counterclockwise from $1$ to $-1$ to $2.$ Thus $\log 2=\ln 2 +i(2\pi).$ We do the same kind of thing from $2$ to $3,$ obtaining $\log 3 = \ln 3 +i(4\pi).$ Now finish by making a counterclockwise half-loop from $3$ to $-3$ to arrive at $\log (-3) = \ln 3 +i(5\pi).$