Is $\log(z^2)=2\log(z)$ if $\text{Log}(z_1 z_2)\ne \text{Log}(z_1)+\text{Log}(z_2)$?

Question

For any $z1, z2$ in $\mathbb{C} \setminus {0}$, $\log(z_1 z_2)=\log(z_1)+\log(z_2)$, but in general $\text{Log}(z_1 z_2)\ne \text{Log}(z_1)+\text{Log}(z_2)$.

Is $\log(z^2)=2\log(z)$?

Answer 1

The answer to the question in the OP is that in general $\displaystyle \log(z^2)\ne 2\log(z)$.

This might seem paradoxical given the relationship expressed as

$$\log(z_1z_2)=\log(z_1)+\log(z_2) \tag1$$

But $(1)$ is interpreted as a set equivalence. It means that any value of $\log(z_1z_2)$ can be expressed as the sum of some value of $\log(z_1)$ and some value of $\log(z_2)$. And conversely, it means that the sum of any value of $\log(z_1)$ and any value of $\log(z_2)$ can be expressed as some value of $\log(z_1z_2)$.

Note that $(1)$ is true since $\log(|z_1z_2|)=\log(|z_1|)+\log(|z_2|)$ and $\arg(z_1z_2)=\arg(z_1)+\arg(z_2)$.

---

EXAMPLE:

As an example, suppose that $z_1=z_2=-1$. Then, $z_1z_2=1$ and $\log(1)=i2n\pi$ for any integer $n$. For $n=0$, $\log(1) =0$. Then $(1)$ is certainly satisfied by $\log(z_1)=i\pi$ and $\log(z_2)=-i\pi$.

Note that although $z_1=z_2$ here, we needed to take two distinct values for $\log(z_1)$ and $\log(z_2)$. We were afforded that degree of freedom since we viewed $z_1$ and $z_2$ as independent.

---

In general,

$$\log(z^2)\ne 2\log(z) \tag 2$$

To see this, we note that for $z=re^{i\theta}$,

$$2\log(z)=2\log(r)+i2(\theta+2n\pi) \tag 3$$

while

$$\begin{align} \log(z^2)&=\log(r^2e^{i2\theta+i2n\pi})\\\\ &=2\log(r)+i2(\theta+n\pi)\tag 4 \end{align}$$

Comparing $(3)$ and $(4)$ we see that $\log(z^2)$ and $2\log(z)$ do not share the same set of values.

---

EXAMPLE:

As an example, suppose $z=i$. For the value $\log(i^2)=i3\pi$, there is no corresponding value of $2\log(z)$. Hence, $\log(i^2)\ne 2\log(i)$ in general.

Answer 2

If $p$ is a non-integer then $z^p$ is a complex multi-valued function and the principal value of $\ln z$ must lie in $- \pi < \Im \ln z< \pi$. From this it follows that \begin{align} \ln (z_1 z_2) &= \ln z_1 + \ln z_2 + 2 \pi i N_{+} \\ \ln \left( z_1 / z_2 \right) &= \ln z_1 - \ln z_2 + 2 \pi i N_{-}\\ \ln z^n &= n \ln z + 2 \pi i N_{n} \end{align} Where $N_{\pm} = 0, +1, -1$ and $$N_n = \frac{1}{2}+\left(\frac{n}{2 \pi}\right)\arg z$$