The logarithms of John Napier (1614) and Henry Briggs (1617) were landmarks in the history of mathematics, and their work is well known to math historians, but not so much to mathematicians who learn calculus from modern texts. This is odd because so many of the basic ideas of analysis are directly related to arithmetic techniques and concepts they employed or anticipated without calculus, and they can all be explained in algebraic terms with limiting values that are evident from calculus. This study explores the underlying mathematics of Napier and Briggs in a manner that underscores modern concepts inherent in their work. Codes written in a Basic-like language are provided to demonstrate the calculations using Casio's online Keisan Calculator.