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Logarithms

Introduction

The logarithm is the power to which a base must be raised to obtain a particular number. That is:

\[ y = b^t \mbox{ iff } t = \log_b y \]

Manipulating Logarithms

The following rules/results make it relatively easy to manipulate logarithms:

  1. \(\log\quad u v = \log u + \log v\)
  2. \(\log u/v = \log u - \log v\)
  3. \(\log u^{k} = k \log u \mbox{ for } u \gt 0\)
  4. \(\log_{b}u = (\log_{b}a) (\log_{a}u) \mbox{ for } u \gt 0\)
    To prove this result let \(u= a^{p}\) (so that \(p=\log_{a}u\)).
    Then: \(\log_{b}u = \log_{b}a^{p} = p\log_{b}a = (\log_{b}a) (\log_{a}u)\)
  5. \(\log_{b}u = \frac{1}{\log_{u}b}\)
  6. Combining the previous two rules yields the following "change of base rule": \(\log_{b}u = \frac{\log_{a}u}{\log_{a}b} \mbox{ for } u \gt 0\)
  7. \(\frac{d}{du} \log_{b}u= \frac{1}{( u \ln b)}\) where \(\ln\) denotes the natural logarithm (i.e., \(\log_{e}\)).

Last modified: 2018-08-26 15:08 by David Bernstein (b3116ea)