## Section2.8Logarithms of Complex Numbers

How can we extend the logarithm function to complex numbers? We would like to retain the property that the logarithm of a product is the sum of the logarithms:

$$\ln(ab)=\ln a+\ln b\label{lnprod}\tag{2.8.1}$$

Then, if we write the complex number $z$ in exponential form:

$$z=r\, e^{i(\theta+2\pi m)}\tag{2.8.2}$$

and use the property (2.8.1), we find:

\begin{align} \ln z\amp = \ln (r\, e^{i(\theta+2\pi m)})\notag\\ \amp = \ln r+ \ln (e^{i(\theta+2\pi m)})\notag\\ \amp = \ln r+ i(\theta+2\pi m)\tag{2.8.3} \end{align}

The logarithm function (for complex numbers) is an example of a multiple-valued function. All of the multiple-values of the logarithm have the same real part $\ln r$ and the imaginary parts all differ by $2\pi\text{.}$

An interesting problem to try is to find $\ln(-1)\text{.}$ You were probably told in high school algebra that the logarithms of negative numbers do not exist.