Logarithms properties, continuity and circular reasoning

Question

I have a doubt relating to logarithms. Anticipating some answers, I know that a "mathy" way of doing things is defining the function by its properties ( $f(xy)=f(x)+f(y)$ ) and then just build up from there. I think this may be a logical way to do it, but makes a disservice to the history of our understanding of both the phenomenon and the math tool to describe it, and may hide a circular reasoning to justify how we define things as we do. Having said that, my question basically reduces to how to build from the ground (first to a high-school student and then to make the connection to a calculus one) the logarithm procedure and the step from discrete to continuous (in the sense of continuous function).

First, as a pedagogical standpoint, I feel the logarithm is the first moment when one can introduce critically the concept of continuity. I will state the problem: The usual presentation is, coming from exponentials, to observe the properties of the logarithm function for the limited cases where the logarithm itself is natural ( $log_2(4*2)=log_2(4)+log_2(2)$ ) and then just apply it to any number. I'm having trouble being both motivating and rigorous when introducing log values that are not natural. For the high-schoolers I would present first the graphic approach and examples in which we may be interested in intermediate values and for the calculus ones, I would first make them prove the properties for rational numbers and then when introducing natural logarithm, I would use that final definition of $b^y$ as $e^{y\ln b}$ .

I would appreciate any references to sources that may make this connection (use of properties as a starting point (given) or following the more historic way of arriving to e as definition after using the properties only for the set o numbers we have proved they work in), or any explanation of a bigger picture connection.

Answer 1

In my opinion, one should forget about trying to be rigorous when introducing logarithms to high school students, or anyone who hasn't seen logarithms before. For students like this, I agree with the answer by Justin Skycak.

If one is trying to be rigorous (say in a college honors calculus or introductory analysis course), there are different ways it can be done. I actually like the presentation that goes the "wrong way around":

1. Define $\ln(x) = \int_1^x (1/t) dt$ , and deduce from this formula all the usual properties of logarithms. In particular, differentiability, and therefore continuity, follow easily from the Fundamental Theorem of Calculus.

2. Define $\exp(x)$ as the inverse function of $\ln(x)$ , and derive the usual properties of $\exp$ from the corresponding properties of $\ln$ . Continuity and differentiability then follow from standard facts about inverse functions.

3. Review the definition of $a^b$ for rational $b$ , and prove that $a^b = \exp(b\ln a)$ whenever $b$ is rational. Motivated by this, define $a^b$ for irrational $b$ as $\exp(b\ln a)$ . Since $\exp$ and $\ln$ are continuous, so is this function.

4. Prove that $\exp(x) = e^x$ for some number $e$ and any real number $x$ .

I call this the "wrong way around" because we think of exponentiation as something we understand better/sooner than logarithms. Certainly, "the inverse of $\ln$ " is not the primary way most people think about the function $e^x$ . But (a) the definition of a thing doesn't have to be the same as the way we think about a thing, and (b) the way I described above is really clean and avoids any potential circularity, since you've defined $\ln x$ for any real positive $x$ before you use it to define $a^b$ for irrational $b$ .

Answer 2

my question basically reduces to how to build from the ground (first to a high-school student and then to make the connection to a calculus one) the logarithm procedure and the step from discrete to continuous (in the sense of continuous function). ... I'm having trouble being both motivating and rigurous when introducing log values that are not natural [i.e., non-integer results].

Personally, I find it easier to introduce students to the idea of fractional exponents, i.e. $a^{b/c} = \sqrt[c]{a^b},$ and then introduce them to the logarithm as the opposite of exponentiation.

If you want to do that rigorously, you can do so by way of inverse functions, which are standard in any high school algebra curriculum.

For high schoolers, I don't think it's important to rigorously extend the treatment to arbitrary real numbers.

On a related note, I've always found that students catch on easier when they are shown an analogy to the relationship between multiplication and division:

• What is $8 \div 2?$ Well, it's the number that you multiply $2$ by to get $8.$

• What is $\log_2 8?$ Well, it's the power of $2$ that gives you $8.$

Answer 3

my question basically reduces to how to build from the ground (first to a high-school student

I agree with Justin Skycak about presenting logarithms as an arithmetic operation by analogy. I think a great example of doing this is on a YouTube video posted about a decade ago by Vi Hart called "How I Feel About Logarithms. " Vi Hart goes back to the number line.

way of arriving to e

My view is that an application orientation in every high school math course except geometry (where proof demands rigor) helps high school students learn math for the general public. The usual math pedagogy where I live seems to introduce e as a useful number to high school students in the application context of continuously compounded interest, and then it's revisited later with more rigor in the calculus context. That seems to work ok.

the logarithm procedure and the step from discrete to continuous (in the sense of continuous function).

First, as a pedagogical standpoint, I feel the logarithm is the first moment when one can introduce critically the concept of continuity.First, as a pedagogical standpoint, I feel the logarithm is the first moment when one can introduce critically the concept of continuity.

Uh oh. I wouldn't trust your feelings about this.

The transition from "a formula for discretely compounded interest" to "a formula for continuously compounded interest" might be the first moment that the potential exists to introduce critical thought about continuity. But I wouldn't do it then, and I wouldn't do it when logarithms are first introduced later or when they are discussed again in high school calculus.

I believe discrete versus continuous function concepts are intuitive to most students in real-world contexts long before high school. The connection between logarithms and continuity seems tenuous to me. When something is intuitive-but-not-rigorous-yet to a student like continuity, I wouldn't mess with it by an example of using something else like logarithms.

I'm having trouble being both motivating and rigorous when introducing log values that are not natural.

I agree with Justin Skycak again. It's enough to discuss rational numbers, but don't worry about real numbers in this context in high school.

Answer 4

Personally, though I haven't had the right class to be able to try this in practice, I'd introduce the logarithm by their (now outdated) practical application. Namely, in a time before calculators, how to you quickly and easily multiply together?

I have a sequence planned where I give the students access to a slide rule and explain the idea (look on the first line for number $a$ , move second line by the corresponding amount, see where $b$ lines up to read off $ab$ ). I then have them fill out a sheet which leads them through the 'some things they might notice about the slide rule' (a.k.a. properties thereof, but mustn't use big words too soon).

Once they have the list of properties, I pose the question of wanting a function that satisfies them, and in front of the class we derive what equalities this function satisfies, e.g. $f(xy) = f(x) f(y)$ but also others.

Here's where I'm a bit fuzzy on what to do next. I think the best way to go around it is to give them some exercice sheets with logarithms until they have familiarity, and only later draw the parallel to exponentiation — namely that $$\text{logarithms}: \times \to +$$ $$ \text{exponents}: + \to \times$$ — and use that to justify the fact that these are inverses of one another. Alternatively, you can note that the properties that $f$ satisfies are those of exponents, and introduce the notion of inverses right away.