The Natural Logarithm of Complex Numbers

Algebra Level 3

Let's extend the natural logarithm to the complex numbers using the definition $\ln{a}=x$ given $e^x=a$ , where $a$ and $x$ can be any complex number. Note that this makes the natural logarithm a multivalued function.

Find the smallest possible value of $|\ln{(3+4i)}|$ .

Bonus: Express $\ln{(a+bi)}$ in algebraic form.

Notations:

$e \approx 2.718$ denotes the Euler's number .
$| \cdot |$ denotes the absolute value function .

The answer is 1.8574624667295145.

1 solution

Chew-Seong Cheong
Nov 16, 2017

Relevant wiki: Euler's Formula

$\begin{aligned} \ln (3+4i) & = \ln \left(5\left({\color{#3D99F6}\frac 35 + \frac 45i}\right)\right) & \small \color{#3D99F6} \text{By Euler's formula: }\cos \theta + i\sin \theta = e^{\theta i} \\ & = \ln 5 + \ln \left({\color{#3D99F6}e^{\left(\tan^{-1}\frac 43 + k\pi\right)i}}\right) & \small \color{#3D99F6} \text{where }k \in \mathbb Z \\ \implies |\ln (3+4i)| & = \sqrt{\ln^25 + \left(\tan^{-1}\frac 43 + k\pi \right)^2} & \small \color{#3D99F6} \text{Minimum when }k=0 \\ & = \sqrt{\ln^25 + \left(\tan^{-1}\frac 43 \right)^2} \\ & \approx \boxed{1.857} \end{aligned}$

Note: some of the identities used here, $\ln{ab}=\ln{a}+\ln{b}$ and $\ln{e^a}=a$ , may not necessarily work with complex numbers.

Nick Turtle - 3 years, 6 months ago

The Natural Logarithm of Complex Numbers

The answer is 1.8574624667295145.

1 solution

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