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Exercises

  1. Show that

    $\displaystyle \frac{d}{dx}\log_b(x) = \frac{1}{x\ln(b)}
$

    where $ \log_b(x)$ denotes the logarithm to the base $ b$ of $ x$ .

  2. Work out the definition of logarithms using a complex base $ b$ .

  3. Try synthesizing a sawtooth waveform which increases by 1/2 dB a few times per second, and again using 1/4 dB increments. See if you agree that quarter-dB increments are ``smooth'' enough for you.


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``Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition'', by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8.
Copyright © 2014-04-06 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
CCRMA