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.

In Polar Form

Where, Amplitude is

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and argument is

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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.

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Therefore,

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

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Examples to clear it all

Find the logarithm of 1 + i?

Step 1: Convert 1 + i into polar form

Now,

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

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Step 3: Take logarithm of both sides

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Or

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Find the logarithm of iota, i

Step 1: Convert i into polar form

Now,

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Step 2: Rewrite iota into exponential form

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Step 3: Take logarithm of both sides.

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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.

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

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