Logarithms: The Dark Sorcery

I used to hate logarithms. They were hopelessly confusing. Sort of like this: https://www.smbc-comics.com/comic/operations

This is the third year now that I’ve been teaching Algebra II. Each year, my understanding of logarithms increases, and my love increases in kind.

One reason I disliked logarithms is because of the way in which we tend to compartmentalize them. It also doesn’t help that we’ve messed up our language about exponents and logarithms.

Here is a mathematical statement involving three variables: $a^b = c$

Variables tend to put people off, so let’s switch to a specific example: $2^3 = 8$

When students first see something like this, we tend to pretend that the “3” position has to be filled with a positive integer. This enables us to then say, “Exponentiation is repeated multiplication.”

It also justifies teaching the radical: With integer-only exponents, the only way to go “backwards” is with roots. Roots, and square roots in particular, are extremely important in mathematics, but they are usually irrational.

In a surely apocryphal story, Pythagoras was so annoyed at the proof for irrational numbers that he threw Hippasus in the sea because of it.

So we introduce radicals and show how they’re the inverse of exponents, and in this process we start to get sloppy about language: $2^3 = 8 \Leftrightarrow \sqrt[3]{8} = 2$

… where we’ll say “Two to the third power is eight” and “the cube root of eight is two”. And we’ll say it so often that students naturally get confused about what the “power” is, and they’ll become teachers later and do the same thing to their students, and eventually “power” will mean the wrong thing.

At some point, we’ll introduce logarithms. And because we’re so used to integers, we’ll hang out with those for the expressions themselves, leaving the irrational numbers for the “result” (which we have no clear name for because… I’ll get to that): $\log_5 12 = 1.5440$

If the teacher is old enough (me, for instance), they’ll remember having to do logs from a table, and so they’ll be used to thinking of $$\log 12 = 1.0792$$ as an exact value unless their teacher back in the day put a lot of effort into pointing out that nearly every number on the log table is a four-digit approximation of an irrational number:

So unless we decided to roll up our sleeves and dig in, or unless we had a college mathematics course where we were forced to roll up our sleeves and dig in, we probably remember logarithms as a painfully mechanical process, vaguely attached to exponential functions but in a way we don’t quite get.

And we teach that to students, who these days seem even more eager to learn things as mechanical processes than we were. Wash. Rinse. Repeat.

Which is a crime, especially these days, when the calculator will gleefully calculate any logarithm you ask it to, store it as an irrational number using whatever magic it uses, and then carry it along through a dozen calculations without adding more than microscopic error.

And it’s a crime these days, when Desmos lets us play with the relationship between exponential functions on x and logarithmic functions on y. While the old faithful erstwhile-godsend now-blight TI-84 can only graph something if you put it in terms of $$y = \text{stuff}$$, Desmos is good with any equation involving two variables.

So:

You can’t see the red graph because it’s under the green graph, which allows us to point out that these are two different ways to write the same relationship. Because logarithmic functions are the inverse of exponential functions.

Yes, the traditional textbooks say that, and then they show two graphs that illustrate that, and students who have trouble visualizing transformations can’t quite grok that they’re inverses of each other:

The relevant point is that we now have some excellent technology with which to play and explore, but so many of us are still stuck in the old ways.

In part, I think, because logarithms were so very painful and mechanical without a calculator, and nearly every mathematics teacher today either learned that way or learned from someone who learned that way.

I’ve complained about language a few times. Let me tell you what the “correct” terms are: $\text{base}^\text{exponent} = \text{power}$

In the example above, “3” is not the power. The power is “8”.

And it’s right there in the language: “The third power of two is eight.” My students are familiar with the language, but they’re also familiar with “Two to the third power is eight”, and when I ask them, they’ll tell me that the power is three.

When I ask other mathematics teachers, they’ll usually tell me that the power is three.

Until I rolled up the sleeves, I thought the power was three.

And here’s why I rolled up my sleeves in the first place. I was told by my Magic Math Book that logarithms are the inverse of exponents, that is, that: $\log_2 8 = 3 \Leftrightarrow 2^3 = 8$

Being the vocabulary-minded teacher that I am, I wanted to call each of these parts something. “Two is the base”. Okay, okay. “Three is the result”. Hm. Not great. But not the worst either. “Eight is the … what?”

So I researched, hoping that I’d find some arcane term. Like “mantissa”, which is the word we might be using instead of referring to the part after the decimal point as “the decimal part”, as if integers aren’t decimals. But that’s a separate rant.

(Properly speaking, before you pedantic at me, the mantissa is the part of the base-10 logarithm after the decimal point, but hey, it’s a reasonable enough expansion.)

That’s how I learned that 8 is the power, which gives us a nice consistent pattern: $\text{base}^\text{exponent} = \text{power} \\ \sqrt[\text{exponent}]{\text{power}} = \text{base} \\ \log_{\text{base}} \text{power} = \text{exponent}$

Here’s our example: $2^3 = 8 \\ \sqrt[3] 8 = 2 \\ \log_2 8 = 3$

This year, I finally wrapped my mind around this simple fact:

We use the logarithm to find the exponent.

Find a place to tattoo that on your brain, and perhaps your Fear of Logarithms will go away, too.

This realization helped something else make sense to me, too. I had been taught that logarithms are the inverse of exponentials, as if exponential functions are the rudimentary thing and logarithms are some sort of subordinate. But then I learned: “Napier first published his work on logarithms in 1614.” And that: “The exponential function arises whenever a quantity grows or decays at a rate proportional to its current value. One such situation is continuously compounded interest, and in fact it was this observation that led Jacob Bernoulli in 1683…”

The in-depth study of logarithms started several generations before the in-depth study of exponential functions. If the sole purpose of logarithms was to find the inverse of an exponential function, why did Napier care in the first place?

This conundrum also explains a different frustration with logarithms: Notation. That nice consistent pattern above represents the same relationship between three values in completely different ways: One with a superscript, one with a wonky symbol, and one with a subscript and a word.

And, for that matter, each one has a different implied value when something’s missing: $2^1 = 2 \\ \sqrt{4} = 2 \\ \log 100 = 2$

A missing exponent is 1; a radical’s missing index is 2; a logarithm’s missing base is 10. Note also that radicals have different language: We use the index (the exponent) and the radicand (the power) to find the root (base).

What a mess.

If you want to see the relationship without the distraction of the inconsistent symbols, 3Blue1Brown offers “The Triangle of Power“.

And if you want to see Vi Hart getting truly excited, here she is talking about logarithms as only she can.

Seeing the base-exponent-power relationship this way allowed me to wrap my mind around another concept. While there are many triads where all values are integers, it is the case for the overwhelming majority of b-e-p triads that at least one is irrational. But it can be any of them, and—here’s the trippy part—we can represent each relationship each way.

Earlier this year, when I was discussing how the implied index for a radical is two and not one, a student asked me if this is a valid thing: $\sqrt[1]5$

That is, what is the first root of five?

My immediate impulse, which I suppressed, was to say that it didn’t exist. But then I thought about it, and asked the student, well, what is the first root of five?

Of course: It’s five. $$\sqrt[1]5 = 5$$ because $$5^1 = 5$$.

But wait… does that mean my radical’s index doesn’t even need to be an integer?

Even the stodgy old TI-84 is good with that:

The only reason why $$\sqrt[0]{5}$$ creates an error is because $$0^x = 5$$ has no solutions $$(\sqrt[0]{0}$$ also returns an error, because $$0^x = 0$$ has infinite solutions).

When using technology, we do still need to retain the rule that a logarithm’s base has to be positive (and not one). It’s not true that $$(-2)^x = 4$$ has no solution, but $$log_{-2} 4$$ returns an error from both Desmos and the TI-84.

But this, I think, is a technical limitation of the way in which those devices calculate logarithms, as well as a traditional restriction, since $$y = (-2)^x$$ is discontinuous and does not have real solutions for most values of $$y$$:

Hence, while real number logarithms with negative bases conceptually exist for relatively few powers, it’s not useful enough to program around the traditional view.

But look: Another strength of modern technology! Rather than merely telling students, “The base of exponential functions and logarithmic functions have to be positive values other than one”, we can readily show what happens when we try negative bases.

Thus ends my rambling for today. I hope you have at least a better appreciation for logarithms than when you started reading.

Clio Corvid