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 numbers4, 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.

  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.↩︎