I typed $\pi$ into Wolfram Alpha and in the short list of definitions there appeared
$ \pi = -i \log(-1)$
which really bothered me. Multiplying on both sides by $2i$:
$ 2\pi i = 2 \log(-1) = \log(-1)^2 = \log 1= 0$
which is clearly false. I guess my error is $\log 1 = 0$ when $\log$ is complex-valued. I need to use $1 = e^{2 \pi i}$ instead.
So my question: is it correct for WA to say $\pi = -i \log(-1)$? Or should be they specifying "which" $-1$ they mean? Clearly $ -1 = e^{i \pi}$ is the "correct" value of $-1$ here.