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., \(\log_{e}\)).