Logarithmic and Complex operations

Assume $w=u+jv=\vert w\vert\angle w=\vert w\vert e^{j\angle w}$, and $z=x+jy=\vert z\vert\angle z=\vert z\vert e^{j\angle z}$.

• 
  
  $$\left\{ \begin{array}{l} \vert w\vert=\sqrt{u^2+v^2},\;\;\;\... ...\;\;\;\angle z=\tan^{-1}\left( y/x \right) \end{array} \right.$$
  
  

• 
  
  $$w z=(u+jv)(x+jy)=\vert w\vert e^{j\angle w}\;\vert z\vert e^{j\angle z}$$
  
  
  
  
  $$\left\vert wz \right\vert =\vert w\vert\vert z\vert,\;\;\;\;\... ...\mbox{or}\;\;\;\; \angle \left( wz \right) =\angle w+\angle z$$
  
  

• 
  
  $$\frac{w}{z}=\frac{u+jv}{x+jy}=\frac{\vert w\vert e^{j\angle w}}{\vert z\vert e^{j\angle z}}$$
  
  
  
  
  $$\left\vert \frac{w}{z} \right\vert =\frac{\vert w\vert}{\vert... ...or}\;\;\; \angle \left( \frac{w}{z} \right)=\angle w-\angle z$$
  
  

• 
  
  $$\log \;(ab)=\log a+\log b$$
  
  

• 
  
  $$\log \;(a/b)=\log a-\log b$$
  
  

• 
  
  $$\log \;(a^n)=n\;\log a$$
  
  

• 
  
  $$\log \;(a^{-n})=-n\;\log a$$