Exercise · ZW-COMPLEX-LOGARITHM-007 · ★★☆ Intermediate
Compute \(\operatorname{Log}\left(e^{4i}\right)\).
Choose one option.
Show answer
- C. \((4-2\pi)\,i\)
Explanation
Since \(4\notin(-\pi,\pi]\), subtract \(2\pi\): \(\operatorname{Arg}(e^{4i})=4-2\pi\approx-2.28\). That is why \(\operatorname{Log}e^{z}\neq z\) in general.
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-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\?