Logs & Complex Numbers

Study Sheet

Logs & Complex Numbers

Change of base, De Moivre, roots of unity

Two Powerful Toolkits

Tip
Change of base — and telescoping logs

log⁡ab=ln⁡bln⁡a\log_a b = \dfrac{\ln b}{\ln a}, so products of logarithms telescope: log⁡ab⋅log⁡bc=log⁡ac\log_a b \cdot \log_b c = \log_a c.

In plain terms. Rewrite every log over one base and watch middle terms cancel.

Example. log⁡23⋅log⁡34=log⁡24=2\log_2 3 \cdot \log_3 4 = \log_2 4 = 2.

Concept
De Moivre's theorem
(cos θ, sin θ)

(cos⁡θ+isin⁡θ)n=cos⁡nθ+isin⁡nθ(\cos\theta + i\sin\theta)^n = \cos n\theta + i\sin n\theta — powers of a unit complex number just multiply the angle.

In plain terms. Raising to a power spins the point around the unit circle nn times as fast.

Example. (cos⁡45∘+isin⁡45∘)2=cos⁡90∘+isin⁡90∘=i(\cos 45^\circ + i\sin 45^\circ)^2 = \cos 90^\circ + i \sin 90^\circ = i.

Tip
Roots of unity sum to zero

The nn solutions of zn=1z^n = 1 are evenly spaced on the unit circle, and their sum is 00 for n≥2n \ge 2.

In plain terms. Evenly spread arrows around a circle cancel out.

Example. For cube roots of unity: 1+ω+ω2=01 + \omega + \omega^2 = 0, which is why ω+ω2=−1\omega + \omega^2 = -1.