# 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. The 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 $\log_a x$. The 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.

## One thought on “Different definitions of logarithm (Part 2)”

This site uses Akismet to reduce spam. Learn how your comment data is processed.