In one line
For \(z=re^{i\theta}\), \(\log z=\log r+i(\theta+2n\pi)\); the principal value takes \(n=0\) and \(\theta\) as the principal argument.
Explained simply
For positive real numbers, \(\log\) answers: "\(e\) to what power gives this number?" For a complex number, the answer is not unique, because \(e^{i\theta}\) repeats every \(2\pi\): going once more round the circle lands at the same point. So a complex number has infinitely many logarithms, differing by \(2\pi i\). We usually pick the one with \(n=0\), called the principal value, written \(\mathrm{Log}\,z\) or just \(\log z\).
Technically, \(z=re^{i(\theta+2n\pi)}\), so \(\log z=\log r+i(\theta+2n\pi)\), \(n\in\mathbb Z\). This also makes sense of the logarithm of a negative number.
How it works
- Write \(z=x+iy\) in polar form: \(r=\sqrt{x^2+y^2}\), \(\theta=\) principal argument (check quadrant).
- General value: \(\log z=\log r+i(\theta+2n\pi)\).
- Principal value: \(\log z=\log r+i\theta\), i.e. \(\dfrac12\log(x^2+y^2)+i\theta\).
- Real part \(=\log|z|\); imaginary part \(=\) argument.
Key formula
\(\log(x+iy)=\dfrac12\log(x^2+y^2)+i\left(\theta+2n\pi\right)\), where \(\theta=\arg(x+iy)\) in \((-\pi,\pi]\) and \(n\) is any integer. For \(x>0\), \(\theta=\tan^{-1}\dfrac yx\).
Useful: \(\log\dfrac{a+ib}{a-ib}=2i\tan^{-1}\dfrac ba\) (principal value, \(a>0\)).
Quick example
- \(\log(-1)\): \(r=1,\ \theta=\pi\). General \(=i(2n+1)\pi\), principal \(=i\pi\).
- \(\log i\): \(r=1,\ \theta=\pi/2\). Principal \(=\dfrac{i\pi}2\).
- \(\log(1+i)\): \(r=\sqrt2,\ \theta=\pi/4\). Principal \(=\dfrac12\log2+\dfrac{i\pi}4\approx0.347+0.785i\).
Check: \(e^{i\pi}=\cos\pi+i\sin\pi=-1\). Correct.
Common mistakes
- Saying \(\log(-1)\) does not exist; it does, as a complex number.
- Using \(\tan^{-1}(y/x)\) as the argument when \(x<0\); fix the quadrant first.
- Applying \(\log(z_1z_2)=\log z_1+\log z_2\) to principal values without care; it holds only up to multiples of \(2\pi i\).
How MU asks it
The syllabus asks for "simple examples", so expect 3-mark parts like "Find the principal value of \(\log(1+i\sqrt3)\)" or 5/6-mark parts like "Prove \(\log\dfrac{a+ib}{a-ib}=2i\tan^{-1}\dfrac ba\)" or "Separate \(\log(1+e^{i\theta})\) into real and imaginary parts". Write the polar form, the general value with \(2n\pi\), and then the principal value.