1.10 Summary Thus Far

The basic rules of logarithms, true for any base \(b\), can be summarized as follows:

  • \(\log_b 1 = 0\)
  • \(\log_b b = 1\)
  • \(\log_b(x \times y) = \log_b x + \log_b y\)

Additionally, from these rules we found that:

  • \(\log_b x^r = r\times \log_b x\)
  • \(\log_b x = \log_a x/\log_a b\)
  • \(\log_b(1/x) = \log_b x^{-1} = - \log_b x\)
  • \(\log_b(x / y) = \log_b x - \log_b y\)

And, by standard convention:

  • Notation when using Base 10: \(\log x \equiv \log_{10} x\)
  • Notation when using Base \(e\): \(~~\;\ln x \equiv \log_e x\)

So far we have discussed the nature of logarithms and their use in performing several types of calculations, assuming that the values of the necessary logarithms have been tabulated. However, though we have talked about a few specific examples, we have not addressed how the value of a logarithm for any arbitrary number can be obtained numerically. The history of calculating logarithms goes back roughly 400 years to the early-1600s, and the techniques of that day are long and laborious. During the following century, after the development of the calculus by Isaac Newton of England and Gottfried Wilhelm Leibniz of Germany, new approaches to such problems made the computation of a logarithm much more tractable. A calculus-based approach is presented in the following chapter from which formulas for directly computing logarithms will be presented.