# If z = 3 - j5, find the value of ln(z).

## Question:

If {eq}z = 3 - j5 {/eq}, find the value of {eq}\ln(z) {/eq}.

## Logarithm of Complex Number:

A logarithmic function can be extended from the set of real into the set of complex numbers. The logarithm of a complex number is a multi-valued function, which is due to the fact that the phase of a complex number is not uniquely defined.

## Answer and Explanation:

First, we will convert the complex number into a polar form:

{eq}z = \sqrt{3^2 + (-5)^2} e^{j\arctan \left (\dfrac {-5}{3} \right)} = \sqrt{34} e^{j\arctan \left (\dfrac {-5}{3} \right)} {/eq}

Calculating the logarithm, we obtain:

{eq}ln (z) = ln \left (\sqrt{34} e^{j\arctan \left (\dfrac {-5}{3} \right)} \right ) = \dfrac {\ln (34)}{2} - \dfrac {5}{3} j + 2\pi n j {/eq}

Here

{eq}n \in \mathbb {Z} {/eq}, we took into account that the phase of a complex number can be defined up to an integer number of {eq}2\pi {/eq}