Complex Logarithm¶

Definition¶

A complex logarithm of a nonzero complex number $z$ is any complex number $w$ for which $\mathrm{e}^{w}=z$$z$. Such a number $w$ is denoted by $\log z$. If $z$ is given in polar form as $z=r\mathrm{e}^{\mathrm{i}\varphi}$$r{\mathrm{e}}^{\mathrm{i}\phi }$, where $r>0$ and $\varphi$ are real numbers, then $\ln r+\mathrm{i} \varphi$ is one logarithm of $z$, and all the complex logarithms of $z$ are exactly the numbers of the form $\ln r+\mathrm{i}\left(\varphi +2\pi k\right)$ for integers $k.$ For each nonzero complex number $z$, the principal value of the complex logarithm $\operatorname{Log} z$ is the logarithm whose imaginary part lies in the interval $(-\pi ,\pi].$

Interpretation¶

A logarithm of $z$ answers the question: to which power must $\mathrm{e}$ be raised to give $z$? For a real $x>0$ there is exactly one answer, the natural logarithm $\ln x$. For a complex $z\ne0$ there are infinitely many, because the complex exponential function is $2\pi\mathrm{i}$-periodic, $\mathrm{e}^{w+2\pi\mathrm{i}}=\mathrm{e}^{w}$${\mathrm{e}}^w$, and therefore not injective.

For $z=r\mathrm{e}^{\mathrm{i}\varphi}$ one has $\mathrm{e}^{\ln r+\mathrm{i}\varphi}=r\mathrm{e}^{\mathrm{i}\varphi}=z$$z$: the real part $\ln r$ of a logarithm records the modulus of $z$, the imaginary part an argument. In the figure all logarithms of $z$ lie on the vertical line with real part $\ln r$ in the $w$-plane, spaced $2\pi$ apart, one for each argument $\varphi+2\pi k$$2\pi k$. The drawn angle $\varphi$ lies in $(-\pi,\pi]$, so the marked middle value $\ln r+\mathrm{i}\varphi$ is $\operatorname{Log}z$; with another choice of argument, $\operatorname{Log}z$ is the value whose imaginary part lies in $(-\pi,\pi]$.

So $\operatorname{Log}z=\ln|z|+\mathrm{i}\operatorname{Arg}z$ is a function, and for a positive real $z$ it is the real $\ln z$. It inherits the jump of $\operatorname{Arg}$: crossing the negative real axis from above to below changes its imaginary part from nearly $\pi$ to nearly $-\pi$. For the same reason, the real rule $\ln(xy)=\ln x+\ln y$ holds for the multivalued $\log$ only as an equality of sets of values; for $\operatorname{Log}$ it can fail by $2\pi\mathrm{i}$, as the Key Results show.

The logarithm is what defines complex powers, $z^w=\mathrm{e}^{w\log z}$${\mathrm{e}}^{w\log z}$, and its many values are the source of the many values of $z^w$.

Key Results¶

• For nonzero $z,w$, the identity $\operatorname{Log}(zw)=\operatorname{Log}z+\operatorname{Log}w$ is not valid in general: the sum of principal arguments may lie outside $(-\pi,\pi]$. For example, taking $z=w=-1$ gives $\operatorname{Log}(zw)=0$$0$, whereas $\operatorname{Log}z+\operatorname{Log}w=2\pi\mathrm{i}$$2\pi \mathrm{i}$. For positive real $z,w$, the identity does hold.

Examples¶

• $\log 1=2\pi\mathrm{i} k$ while $\operatorname{Log} 1=0.$
• $\log (-1)=\mathrm{i} (\pi+2\pi k)$ while $\operatorname{Log} (-1)=\mathrm{i}\pi.$

Prerequisites¶

• Complex Exponential Function
• Polar Representation of Complex Numbers
• Complex Number

Related Concepts¶

• Argument of a Complex Number
• Euler's Formula
• Complex Exponentiation
• Logarithm and Logarithmic Function

Sources¶

• Math II: Linear Algebra · Week 1

Aliases: logarithm of a complex number Symbols: phi, pi

Last updated: 27 October 2026 11:52