**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.