# Thread: Any two, non zero, complex numbers.

1. ## Any two, non zero, complex numbers.

Log is the principal branch of the log function.
If two non-zero complex numbers $\displaystyle z_{1},z_{2}$
Then show the following:
Log($\displaystyle z_{1}z_{2}$) = Log($\displaystyle z_{1}$) + Log($\displaystyle z_{2}$) - 2N*pi*i, for N = 0 or N = 1

2. Originally Posted by universalsandbox
Log is the principal branch of the log function.
If two non-zero complex numbers $\displaystyle z_{1},z_{2}$
Then show the following:
Log($\displaystyle z_{1}z_{2}$) = Log($\displaystyle z_{1}$) + Log($\displaystyle z_{2}$) - 2N*pi*i, for N = 0 or N = 1

Let $\displaystyle z_1 = r_1e^{i\theta_1},z_2=r_2e^{i\theta_2}$ where $\displaystyle \theta_1,\theta_2 \in (-\pi,\pi]$.
Then, $\displaystyle z_1z_2 = r_1r_2 e^{i(\theta_1+\theta_2)}$
Therefore, $\displaystyle \arg (z_1z_2) = \theta_1 + \theta_2 + 2\pi N$ where $\displaystyle N =0\text{ or }-1 \text{ or }1$.
But, $\displaystyle \log z_1 = \ln r_1 + i \theta_1 \text{ and }\log_2 = \ln z_2 + i \theta_2$.
And, $\displaystyle \log z_1z_2 = \ln r_1r_2 + i (\theta_1+\theta_2) + 2i\pi N$
From here it follows that $\displaystyle \log z_1z_2 = \log z_1 + \log z_2 + 2\pi i N$