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Welcome to our Math lesson on The History of Logarithm, this is the second lesson of our suite of math lessons covering the topic of Definition and Properties of Logarithms, you can find links to the other lessons within this tutorial and access additional Math learning resources below this lesson.
The invention of logarithms was cued by the need to find a connection between arithmetic and geometric sequences. From the previous chapter, it is known that in a geometric sequence each term forms a constant ratio with the next one. For example, the geometric sequence
has a common ratio R = 10, as the terms increase by a factor of 10 with respect to the previous one. On the other hand, in an arithmetic sequence, each successive term differs by a constant from the previous one. This constant is known as the common difference d. For example, the arithmetic series
has a common difference of d = 1. Note that a geometric sequence can be written in terms of its common ratio; for the example, the geometric sequence given above can be written as:
Multiplying two numbers in the geometric sequence, for example 1/10 and 100, is equal to adding the corresponding exponents of the common ratio, −1 and 2. In both cases, we obtain 101 = 10. In this way, multiplication is transformed into addition. The original comparison between the two series, however, was not based on any explicit use of the exponential notation; this was a later development. The first table based on the concept of relating geometric and arithmetic sequences was published in 1620.
Based on the above findings the concept of the logarithm was defined - a concept which was first proposed by John Napier. Thus, instead of considering the geometric sequence
we can consider only their logarithm. In this way, we obtain the corresponding arithmetic series
given that
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