I tried to draw a spiral $y=(a+bi)^x$ in a complex plane, but I found the differences. The method I used was:
\begin{align} & a+bi = (a, b). \\ & r_0=r_{a, b}=\sqrt{a^2+b^2}, \theta_0=\theta_{a, b}=\arctan\left(\frac b a\right). \\ & \text{Note. } r_{(a, b) \cdot (c, d)}=r_{a, b} \cdot r_{c, d}, \theta_{(a, b) \cdot (c, d)} = \theta_{a, b} + \theta_{c, d}. \\ \Rightarrow \; & \text{let } (a+bi)^x = (a_x, b_x), r_{a_x, b_x} = (r_{a, b})^x, \theta_{a_x, b_x} = x \cdot \theta_{a, b}. \\ \ \\ & r = (r_0)^{\frac {\theta}{\theta_0}} \tag{1}\label{(1)} \\ & y=x\cdot \tan \left( \left( \log_{a^2+b^2} (x^2+y^2) \right) \cdot \arctan \left( \frac b a \right) \right) \tag{2}\label{(2)} \\ \end{align}
By drawing these two graphs, the plane has two DIFFERENT charts:
Is it because of the degree and the radian? Can anyone explain this...