Logarithms of Negative and Imaginary Numbers
By Euler's identity,
, so that
![$\displaystyle \ln(-1) = j\pi
$](http://www.dsprelated.com/josimages_new/mdft/img1922.png)
![$ x<0$](http://www.dsprelated.com/josimages_new/mdft/img1923.png)
![$ \ln(x) = j\pi + \ln(\vert x\vert)$](http://www.dsprelated.com/josimages_new/mdft/img1924.png)
Similarly,
, so that
![$\displaystyle \ln(j) = j\frac{\pi}{2}
$](http://www.dsprelated.com/josimages_new/mdft/img1926.png)
![$ z = jy$](http://www.dsprelated.com/josimages_new/mdft/img1927.png)
![$ \ln(z) = j\pi/2 + \ln(y)$](http://www.dsprelated.com/josimages_new/mdft/img1928.png)
![$ y$](http://www.dsprelated.com/josimages_new/mdft/img26.png)
Finally, from the polar representation
for
complex numbers,
![$\displaystyle \ln(z) \isdef \ln(r e^{j\theta}) = \ln(r) + j\theta
$](http://www.dsprelated.com/josimages_new/mdft/img1929.png)
![$ r>0$](http://www.dsprelated.com/josimages_new/mdft/img1930.png)
![$ \theta$](http://www.dsprelated.com/josimages_new/mdft/img182.png)
![$ e^{j\theta }$](http://www.dsprelated.com/josimages_new/mdft/img3.png)
![$ j$](http://www.dsprelated.com/josimages_new/mdft/img89.png)
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Properties of DB Scales
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