Exercise · ZW-COMPLEX-LOGARITHM-006 · ★★☆ Intermediate
Compute \(\operatorname{Log}(-2i)\).
Choose one option.
Show answer
- A. \(\ln 2-\tfrac{\pi}{2}i\)
Explanation
\(|-2i|=2\) and \(\operatorname{Arg}(-2i)=-\tfrac{\pi}{2}\); \(\tfrac{3\pi}{2}\) is a valid argument but not the principal one.
Review the concept: Complex logarithm
A logarithm of a complex number \(z\neq0\) is a complex number \(w\) whose exponential is \(z\).
\[\log z=\ln|z|+i(\arg z+2\pi k),\quad k\in\mathbb{Z}\]
More practice problems: complex logarithm
- ZW-COMPLEX-LOGARITHM-001 Compute the principal logarithm Log i.
- ZW-COMPLEX-LOGARITHM-002 Compute Log(1+i).
- ZW-COMPLEX-LOGARITHM-003 Compute Log(-1).
- ZW-COMPLEX-LOGARITHM-004 What are all the logarithms of 1 (with k∈ℤ)?
- ZW-COMPLEX-LOGARITHM-005 Compute Log(-e).
- ZW-COMPLEX-LOGARITHM-007 Compute Log(e^4i).
- ZW-COMPLEX-LOGARITHM-008 With z=w=-1, compute Log(zw)-Logz-Logw.
- ZW-COMPLEX-LOGARITHM-009 At which points is the principal logarithm Log discontinuous within its domain ℂsetminus\0\?