Different definitions of logarithm (Part 1)

There are two apparently different definitions of a logarithm that appear in the secondary mathematics curriculum:

  1. From Algebra II and Precalculus: If b > 0 and b \ne 1, then f(x) = \log_b x is the inverse function of g(x) = b^x.
  2. From Calculus: for x > 0, we define \ln x = \displaystyle \int_1^x \frac{1}{t} dt.

On the surface, these two ways of viewing logarithms are completely separate from each other, and so even advanced math majors are surprised that these two ways of viewing logarithms are logically interrelated. In the words of Tom Apostol (Calculus, Vol. 1, 2nd edition, 1967, page 227):

The logarithm is an example of a mathematical concept that can be defined in many different ways. When a mathematician tries to formulate a definition of a concept, such as the logarithm, he usually has in mind a number of properties he wants this concept to have. By examining these properties, he is often led to a simple formula or process that might serve as a definition from which all the desired properties spring forth as logical deductions.

In this series of posts, we examine the interrelationship between these two different approaches to logarithms. This is a standard topic in my class for future teachers of secondary mathematics as a way of deepening their understanding of a topic that they think they know quite well.


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