Exercise · ZW-COMPLEX-LOGARITHM-008 · ★★☆ Intermediate
With \(z=w=-1\), compute \(\operatorname{Log}(zw)-\operatorname{Log}z-\operatorname{Log}w\).
Choose one option.
Show answer
- D. \(-2\pi i\)
Explanation
\(\operatorname{Log}1=0\), but \(\operatorname{Log}(-1)=\pi i\), so the difference is \(0-2\pi i=-2\pi i\). The product rule only holds up to multiples of \(2\pi i\).
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-009 At which points is the principal logarithm Log discontinuous within its domain ℂsetminus\0\?