Question

No matter how hard I try, I can't seem to get this right! Please help! Q: Give an exp-log expression for (−√3−i)^i.

Answer 1

\[y=(−√3−i)^i\qquad\rightarrow\qquad\log_{(−√3−i)}y=i\]
\[-i\log_{(−√3−i)}y=1\]
\[\log_{(−√3−i)}y^{-i}=1\]
\[\log_{(−√3−i)}\frac{1}{y^{i}}=1\]

Answer 2

\[z^i=e^{i\log(z)}\]

Answer 3

and
\[\log(z)=\ln(z)+i\arg{z}\]

Answer 4

\[-\sqrt{3}-i=2e^{\frac{7\pi}{6}}\]and so
\[i\log(z)=\ln(2)+i\frac{7\pi}{6}\]