$\\Log$ denotes the multi-valued natural log defined over the complex numbers
I know that $Log(z)$ is multivalued and that we can choose a particular branch if we want to, but why isn't it the case that $Log(-1)=Log(1/(-1))=-Log(-1) ⟹Log(-1)=0⟹(\theta/\pi)Log(e^{i\pi})=Log(e^{i\theta})=0⟹Log(z)=\ln(|z|)$
Choosing the principal branch, $i\pi=Log(-1)=Log(1/(-1))=-Log(-1)=-i\pi$ . I've never understood why we're allowed to choose a branch even though a contradiction like this can be derived.
Edit: I've understood what I was misunderstanding before. Of course, the rules for the single valued logarithm are not applicable for numbers that are not an element of the positive reals. My mistake
The principal branch logarithm is not a homomorphism: in general $\log(zw) \not= \log z + \log w$ for nonzero complex numbers $z$ and $w$ off the negative real axis, so all of your experience about "rules" of logarithms is generally wrong when you start using numbers outside the positive real axis inside the logarithm function. That means the calculations in your post are mostly nonsense.
If you make incorrect calculations, then it's not surprising that you are reaching "contradictions". There is in fact no contradiction because your premises are just not true. The error you are making is due to implicitly treating the valid equation $\log(xy) = \log x + \log y$ for positive real $x$ and $y$ as if it holds in general for nonzero complex $x$ and $y$, which is a mistake.
We learn in school that for positive real numbers, $a^2 = b^2 \Longrightarrow a = b$. If someone then comes along and says "what about $3^2 = (-3)^2 \Longrightarrow 3 = -3$?" Well, that's just an incorrect conclusion since it is based on a property that nobody ever said was true about nonzero real numbers; it true for positive real numbers, and if your go ahead and use it in a wider setting without checking whether it in fact holds there, you could potentially get nonsense, which is indeed what happens.