Exercise · ZW-COMPLEX-LOGARITHM-010 · ★★★ Advanced
Find all logarithms of \(-2\) (with \(k\in\mathbb{Z}\)).
Choose one option.
Show answer
- A. \(\ln 2+(2k+1)\pi i\)
Explanation
\(|-2|=2\) and the arguments of \(-2\) are \(\pi+2\pi k=(2k+1)\pi\). With \(2k\pi\) you would get \(e^{w}=2\), and \(\ln(-2)\) is undefined in \(\mathbb{R}\).
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-006 Compute Log(-2i).
- ZW-COMPLEX-LOGARITHM-007 Compute Log(e^4i).
- ZW-COMPLEX-LOGARITHM-008 With z=w=-1, compute Log(zw)-Logz-Logw.