Different definitions of logarithm (Part 2)

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.

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.

green lineThe connection between these two apparently different ideas begins with the following theorem.

Theorem. Let a \in \mathbb{R}^+ \setminus \{1\}. Suppose that f: \mathbb{R}^+ \rightarrow \mathbb{R} has the following four properties:

  1. f(1) = 0
  2. f(a) = 1
  3. f(xy) = f(x) + f(y) for all x, y \in \mathbb{R}^+
  4. f is continuous

Then f(x) = \underline{\hspace{1in}} for all x \in \mathbb{R}^+.

When writing this on the board, I purposefully leave an underline for my students to fill in, because I want them to think. What familiar function has these four properties? I’ll usually invoke the old chidren’s joke: “If it looks like an elephant, smells like an elephant, feels like an elephant, and sounds like an elephant, then it must be an elephant.” After a moment of thought, someone will usually volunteer f(x) = \log x. That’s almost correct, and so I’ll ask if Property 2 is satisfied by this function. After a couple more moments of thought, someone will volunteer the correct answer, f(x) = \log_a x.

To prove this theorem, I will show that

f(a^x) = x for all x \in \mathbb{R}.

I’ll make the observation that the case of $latex  x=0$ is Property 1, while the case of x = 1 is Property 2.

Then I’ll ask the class: “If I’m able to prove that f(a^x) = x for all real x, why does this mean that f(x) = \log_a x?” Perhaps unsurprisingly, this usually draws blank stares for a few seconds until someone realizes that this means that f: \mathbb{R}^+ \rightarrow \mathbb{R} and g: \mathbb{R} \rightarrow \mathbb{R}^+ defined by g(x) = a^x are inverse functions. So (by definition) f(x) must be equal to \log_a x.

green lineThe proof of these theorem has four parts:

  1. Positive integers: x = m \in \mathbb{Z}^+
  2. Positive rational numbers: x = \frac{m}{n}, where m,n \in \mathbb{Z}^+
  3. Negative rational numbers: x \in \mathbb{Q}^-
  4. Real (possibly irrational) numbers: x \in \mathbb{R}

Beginning with tomorrow’s post, I’ll discuss how I walk students through the proof in lecture.

 

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