The Logarithmic Function

---

Consider $z$ any nonzero complex number. We would like to solve for $w,$ the equation \begin{eqnarray}\label{log1} e^w=z. \end{eqnarray} If $\Theta=\textbf{Arg} (z)$ with $-\pi < \Theta \leq \pi,$ then $z$ and $w$ can be written as follows \begin{eqnarray*} z=re^{i\Theta} \quad \text{and}\quad w=u+iv. \end{eqnarray*} Then equation (\ref{log1}) becomes $$e^ue^{iv}=re^{i\Theta}.$$ Thus, we have \begin{eqnarray*} e^u=r\quad\text{and}\quad v=\Theta +2n\pi \end{eqnarray*} where $n\in \mathbb Z .$ Since $e^u=r$ is the same as $u=\ln r,$ it follows that equation (\ref{log1}) is satisfied if and only if $w$ has one of the values

Therefore, the (multiple-valued) logarithmic function of a nonzero complex variable $z=re^{i\Theta}$ is defined by the formula

The principal value of $\log z$ is the value obtained from equation (\ref{log2}) when $n = 0$ and is denoted by $\text{Log} \,z.$ Thus $$\text{Log}\, z=\ln r +i\Theta .$$ The function $\text{Log}\, z$ is well defined and single-valued when $z\neq 0$ and that $$\log z=\text{Log}\, z+2n\pi i\quad (n\in \mathbb Z ) $$ This is reduced to the usual logarithm in calculus when $z$ is a positive real number.

Expression (\ref{log2}) is also equivalent to the following: \begin{eqnarray*} \log z&=&\ln |z| +i\,\textbf{arg} (z)\\ &=&\ln |z| +i\,\textbf{Arg} (z)+2ni\,\pi \quad \quad(n\in \mathbb Z ) \end{eqnarray*}

Some basic properties of the function $\log z$ are the following:

1. $\log \left(z_1 \,z_2\right)=\log z_1 + \log z_2$
2. $\log\left( \dfrac{z_1}{z_2}\right)=\log z_1 -\log z_2$
3. There may hold $\text{Log}\,\left(z_1 \,z_2\right)\neq \text{Log}\, z_1 + \text{Log}\, z_2$

Branches of logarithms

From definition (\ref{log2}) let $\theta = \Theta + 2n\pi$ ($n\in \mathbb Z$), so we can write \begin{eqnarray}\label{log30} \log z = \ln r +i\theta. \end{eqnarray}

Now, let $\alpha$ be any real number. If we restrict the value of $\theta$ so that $\alpha < \theta < \alpha + 2n\pi$ , then the function

with components \begin{eqnarray*} u(r, \theta)=\ln r, \quad v(r,\theta)=\theta, \end{eqnarray*} is a single-value and continuous function in the stated domain.

A branch of a multiple-valued function $f$ is any single-valued function $F$ that is analytic in some domain at each point $z$ of which the value $F(z)$ is one of the values of $f.$ The requirement of analyticity, of course, prevents $F$ from taking on a random selection of the values of $f.$ Observe that for each fixed $\alpha,$ the single-valued function (\ref{log3}) is a branch of the multiple-valued function (\ref{log30}). The function

is called the principal branch.

A branch cut is a portion of a line or curve that is introduced in order to define a branch $F$ of a multiple-valued function $f.$ Points on the branch cut for $F$ are singular points of $F,$ and any point that is common to all branch cuts of $f$ is called a branch point. The origin and the ray $\theta = \alpha$ make up the branch cut for the branch (\ref{log3}) of the logarithmic function. The branch cut for the principal branch (\ref{log4}) consists of the origin and the ray $= \pi.$ The origin is evidently a branch point for branches of the multiple-valued logarithmic function.

We can visualize the multiple-valued nature of $\log z$ by using Riemann surfaces. The following interactive images show the real and imaginary components of $\log(z).$

Special care must be taken in using branches of the logarithmic function, especially since expected identities involving logarithms do not always carry over from calculus.

Final remark

Notice that for $z\neq 0,$ we have

with $n\in \mathbb Z .$