As many of us have been taught, a logarithmic function in the inverse of an exponential function. Based on this concept, some text even introduce logarithms by saying a “A logarithm is an exponent”. (Yoshiwara & Yoshiwara). Often times we say that a logarithmic function log_b(x)=y means that an x=b^y. A logarithm means we are looking for the power on b that would produce the value x. Or, in more precise words: “A logarithm is an exponent; it is the exponent to which b must be raised to yield x” (Yoshiwara & Yoshiwara).

Before doing the research for this website, I thought that this was all a logarithmic function was. However, I have learned this is only a small part of the mathematics behind a logarithm. Also, this wasn't the original purpose of a logarithm.

Benjamin Martin argued that Edward Wright was the first to invent a system of logarithms. The document Martin was referring to was Wright's table for navigation that compared numbers that increased linearly with numbers that increased proportionally. It is said that “Wright ‘happened upon the logarithms and he did not know it' ” (Knott, 1915). How did this happen? “That such a table should turn out to be a table of logarithm is not as strange as it might seem. Any set of numbers in arithmetical progression placed parallel with a set of positive numbers in geometrical progression defines some system of logarithms” (Knott, 1915). So any two lists of numbers, one with a set difference between the values and the other with a multiple difference between the values, would produce something that resembled a logarithmic relationship.

Imagine two number lines, the first with the counting numbers in order, the other a geometric progression with some simple base, say 3 (see diagram below). When relating these two sets of numbers, wouldn't we say that the first would represent the exponent on the base of the second? Or in other words, if we were to relate this to logarithms, if you were to take a logarithm of that same base of the second set of numbers, would it not produce the first?

This is why logarithms were invented.

An interesting demonstration of the above concept is found in this lesson plan.