Changing the Base
By definition,
. Taking the log base
of both sides
gives
![$\displaystyle \log_a(x) = \log_b(x) \log_a(b)
$](http://www.dsprelated.com/josimages_new/mdft/img1920.png)
![$ b$](http://www.dsprelated.com/josimages_new/mdft/img818.png)
![$ a$](http://www.dsprelated.com/josimages_new/mdft/img236.png)
![$ b$](http://www.dsprelated.com/josimages_new/mdft/img818.png)
![$ x$](http://www.dsprelated.com/josimages_new/mdft/img25.png)
![$ a$](http://www.dsprelated.com/josimages_new/mdft/img236.png)
![$ x$](http://www.dsprelated.com/josimages_new/mdft/img25.png)
![$ a$](http://www.dsprelated.com/josimages_new/mdft/img236.png)
![$ b$](http://www.dsprelated.com/josimages_new/mdft/img818.png)
Next Section:
Logarithms of Negative and Imaginary Numbers
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Continuous-Time Aliasing Theorem
By definition,
. Taking the log base
of both sides
gives