Show that
- \(\log(zw)\neq \log(z)\log(w)\)
- \(\log(\exp(z))\neq z\)
Solution
Counterexamples:
Take \(z=\zeta_4^3\) and \(w=\zeta_4^2\), noting that \(\Arg(z) = {3\pi \over 4}\) and \(\Arg(w) = {\pi \over 2}\) Then \(zw = e^{5\pi \over 4}\) but \(\Arg(zw) = {-3\pi \over 4}\) because we are forced to use the domain \((-\pi, \pi]\).
Showing that \(\log(\exp(z))\) is a countably infinite set in \(\CC\): \[ \log(\exp(z)) &= \log(\exp(x+iy)) \\ &= \ln\qty{e^x} + (y+2k\pi)i \\ &= (x+iy) + 2k\pi i \\ &= z + 2k\pi .\]