There are two apparently different definitions of a logarithm that appear in the secondary mathematics curriculum:
- From Algebra II and Precalculus: If and , then is the inverse function of .
- From Calculus: for , we define .
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.
Theorem. Let . Suppose that has the following four properties:
- for all
- is continuous
Then for all .
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 . 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, .
To prove this theorem, I will show that
for all .
I’ll make the observation that the case of $latex x=0$ is Property 1, while the case of is Property 2.
Then I’ll ask the class: “If I’m able to prove that for all real , why does this mean that ?” Perhaps unsurprisingly, this usually draws blank stares for a few seconds until someone realizes that this means that and defined by are inverse functions. So (by definition) must be equal to .
- Positive integers:
- Positive rational numbers: , where
- Negative rational numbers:
- Real (possibly irrational) numbers:
Beginning with tomorrow’s post, I’ll discuss how I walk students through the proof in lecture.