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: \begin{equation} \ln(ab)=\ln a+\ln b \label{lnprod} \end{equation} Then, if we write the complex number $z$ in exponential form: \begin{equation} z=r\, e^{i(\theta+2\pi m)} \end{equation} and use the property (\ref{lnprod}), we find: \begin{eqnarray*} \ln z&=&\ln (r\, e^{i(\theta+2\pi m)})\\ &=&\ln r+ \ln (e^{i(\theta+2\pi m)})\\ &=&\ln r+ i(\theta+2\pi m) \end{eqnarray*} 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 $2\pi$.

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