Complex logarithms, the principal logarithm, and principal and multivalued complex powers

For $z\ne0$ with principal polar form $z=r(\cos\theta+i\sin\theta)$, $-\pi<\theta\le\pi$, define the principal logarithm $\operatorname{Log}z:=\log r+i\theta.$ The set of all complex logarithms of $z$ is $\log_{\mathrm{multi}}z:=\{v\in\mathbb C:\exp v=z\}.$ For $w\in\mathbb C$, define the principal power and the multivalued power respectively by $z^w_{\mathrm{pr}}:=\exp(w\operatorname{Log}z),\qquad z^w_{\mathrm{multi}}:=\{\exp(wv):v\in\log_{\mathrm{multi}}z\}.$ Thus the first is one specified complex number and the second is a set of complex numbers; neither notation silently identifies them. The conventions and prerequisite facts used below are recorded in Every nonzero complex number has a unique polar form $r(\cos\theta+i\sin\theta)$ with $r>0$ and $-\pi<\theta\le\pi$, The natural logarithm as the inverse of the exponential function, The complex exponential by its power series, The complex numbers form a field, and every nonzero $x+iy$ has inverse $(x-iy)/(x^2+y^2)$.