How to find the logarithm of a complex number?

Table Of Content

Brush Up Basics

Let a + ib be a complex number whose logarithm is to be found.

Step 1: Convert the given complex number, into polar form.

Where, Amplitude is

and argument is

Step 2: Use Euler’s Theorem to rewrite complex number in polar form to exponential form.

There r (cos θ + i sin θ) is written as reiθ. This means that

a+ ib = reiθ

Step 3: Take logarithm of both sides we get.

Therefore,

The above results can be expressed in terms of modulus and argument of z.

Examples to clear it all

Find the logarithm of 1 + i?

Step 1: Convert 1 + i into polar form

Now,

Step 2: Use Euler’s Theorem to rewrite complex number

Step 3: Take logarithm of both sides

Or

Find the logarithm of iota, i

Step 1: Convert i into polar form

Now,

Step 2: Rewrite iota into exponential form

Step 3: Take logarithm of both sides.

Observations to give you insight

Why should we convert a complex number to its exponential form? The answer is simple: Exponentials are also known as anti-logarithms. It requires no brilliance that taking log of anti-log gives us the log of that number.

The logarithm of a complex number can be a real number only if

Argument of a complex number can only be zero if its imaginary part, b is zero.