2 Computing Logarithms

In the last chapter we gave examples of how one could fill in a curve of the form \(x=b^p\) in a laborious iterative fashion and, by inverting the result, one could in principle obtain the exponent \(p\) (the logarithm) that corresponded to a given \(x\) for that particular base. Rather than taking such an iterative approach and “filling in” tables of numbers5, we would rather be able to generate a formula for a logarithm of any given number (and, in fact, for any chosen base) and be able to compute it to any desired accuracy.

In what follows we will use calculus to find a natural base to use for our computations. With our appropriate definition of a natural logarithm we can use a standard technique to find a Taylor Series in terms of the argument \(x\) to create a formula for computing the natural logarithm of \(x\) using our natural base. Then, by using one of our general rules of logarithms, we can find the common (Base 10) logarithm of the number \(x\) to any reasonably desired accuracy. Those readers not versed in the mathematics of the calculus should not be deterred from reading through the following sections. Equations for computing logarithms, including example lines of computer code, are presented which might still be of interest to some readers. So even though the details presented might appear esoteric to the non-expert, they are provided for completeness of the topic of the logarithm.

  1. It is estimated that Napier spent 20 years developing his first complete table of logarithms! And the connection between Napier’s approach and an “exponential” approach would not be made for yet another 20 years after that.↩︎