Logarithms: The Early History of a Familiar Function - Conclusion

Logarithms represented at this time in so many ways both what was old and what was new. This relation looked back to reflect concerns of computation, but looked forward to nascent notions about mathematical functions. Although logarithms were primarily a tool for facilitating computation, they were but another of the crucial insights that directed the attention of mathematical scholars towards more abstract organizing notions. But one thing is very clear: the concept of logarithm as we understand it today as a function is quite different in many respects from how it was originally conceived. But eventually, through the work, consideration, and development of many mathematicians, the logarithm became far more than a useful way to compute with large unwieldy numbers. It became a mathematical relation and function in its own right.

In time, the logarithm evolved from a labor saving device to become one of the core functions in mathematics. Today, it has been extended to negative and complex numbers and it is vital in many modern branches of mathematics. It has an important role in group theory and is key to calculus, with its straightforward derivatives and its appearance in the solutions to various integrals. Logarithms form the basis of the Richter scale and the measure of pH, and they characterize the music intervals in the octave, to name but a few applications. Ironically, the logarithm still serves as a labor saving device of sorts, but not for the benefit of human effort! It is often used by computers to approximate certain operations that would be too costly, in terms of computer power, to evaluate directly, particularly those of the form \(x^n\).