# Laying the Groundwork for Logarithms

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Strangely, I have had occasion to do a few tutoring sessions with different kids recently around exponential and logarithmic functions.

This particular mistake set off a few alarm bells:

Do you see what the student is doing here? She is treating

like a variable that is being divided instead of a function.

I looked at the student’s notes, and all the usual log laws were there. But she did not yet have the unshakable understanding that logs are functions. I realized that there are some foundational ideas that she needed before we could really make sense of all of this.

Here are a couple of essential ideas I want to communicate to students about logarithm functions.

First, functions can be described as actions, so I always make students explain what a function is doing.

The question you should ask about every function is: what are we doing to the input to get to the output? I call it “saying the function in English.”

Since we usually teach logarithms after exponential functions, let me start with them.

I ask, What do exponential functions do? They provide rules based on repeated multiplication. So the function

tells us that “some number (y) equals 2 multiplied by itself any number of (x)  times” to get y. We can do this with different examples, talk about how the function grows, look at the graph, look at tables, compare the growth of exponential functions to linear and quadratic ones. My goal is to get kids to have a feel for what is happening with exponential growth so well that when somebody says, “It was growing exponentially!” they can decide whether that is an accurate statement or not.

This is the first part of the groundwork for understanding logarithms.

Second, remember that anything we do in mathematics, we always find ways to undo.

This is thematic in all of mathematics. It becomes a chant when I teach math.
I say to students:
“Since this is math, anything we learn to do, we need to ….?”
They soon learn to respond with:
UNDO!!!”

Doing-and-undoing is a good mathematical habit of mind to emphasize, because students start to anticipate that when we learn some new funky function or operation, an inverse is coming down the pike. They are not at all surprised to learn that trig functions have an inverse and so on.

In this case, since we have learned to exponentiate, they can guess we need to un-exponentiate.

That’s just how math works!

I like to show inverses of functions in all of the representations. The idea is the same in tables, graphs and equations: the x’s and y’s switch places.

For tables and graphs, it’s fairly easy for students to figure it out. But the algebra gets tricky. To find the inverse of the previous exponential, for example, we need to derive it from:

This immediately creates a mathematical need to “un-exponentiate.”

So when we want to solve that equation for y, let’s undo exponentiation with a function we call a logarithm. Logarithms undo exponentiation.

Since the log undoes the exponentiation, we end up isolating the y.

I also tell them we read this as “log base two of x equals y.”

So when you see an equation like:

you are asking “2 to what power equals 8?” I have them practice explaining what different equations mean.

Now your students are ready to learn all the details of working with logs!

[Before I close, vaguely related Arrested Development reference:

Because this is a log law blog. But I guess I don’t really want to talk about log laws. Anyways…]

## 25 thoughts on “Laying the Groundwork for Logarithms”

1. “We’re not here to talk nonsense to Bob Loblaw…”

Great post.

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2. Our Alg2 textbook spends several days calling logarithms the mysterious L(x), with the option to include a base such as L_2(x) for base 2. This attempts to solidify the functional details, and forces kids to use the inverse property. A lot of these errors happen because kids don’t see log as a function — the notation used with no parentheses also supports this.

If nothing else, I recommend writing log (1000) instead of log 1000, which helps emphasize the fact that we’re dealing with functions. I feel the same way about trigonometric functions, that many of the errors we see are caused by a misunderstanding of the functional nature of these objects.

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• notation causes so many problems. i like the L(x) idea a lot.

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• Will have to post some examples of this to the Patterns in Practice blog. Time!

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3. I also like having students use their own “verb-language” when applying an exponential function, logarithmic function, or trigonometric function: Sometimes the results are a bit awkward when the share their words – “2-to-the input,” “logarithmize,” “de-exponentiate,” but it helps them reinforce the fact that they are functions, and they like making up their own mathematical language that is in spirit, on point, but sounds a bit silly at first… Once they get the point, then I opportunity to share with them more commonly used items (“exponentiate base 2,” “take the logarithm-base-2 of…”).

I also think that we can often prematurely drop the use of parentheses when using logarithmic functions… We tend to write “log xy” when “log(xy)” might be a subtle but helpful reminder. Eventually they have to see both as the same expression, but sometimes this equivalence gets obscured when they start solving equations expressions that involve both algebraic properties and properties of exponential and logarithmic functions.

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4. This is (by far) the most common error that my students make when dealing with logs. It’s really no surprise, I suppose. One of the first things that students learn is that if log a = log b, then a = b. It sure looks like we’re canceling here, right? It’s essential to remind students at this critical point that we are NOT canceling here! (In fact, I’d love to go back to pre-Algebra classes and remove canceling from our vocabulary. Change it to undoing the multiplication, since that’s really what we are doing. But that’s a separate issue, isn’t it?)

I’ve got to agree as well with earlier comments that writing a logarithm with parentheses as log (x) instead of log x reinforces the idea that it’s a function, and really has to be undone somehow.

Even so, with all the tricks that I’ve learned over the years about how to teach logs, some students simply have so much anxiety during a test that they make mistakes like this, even when they haven’t made it during the entire unit, because the stress leads them to fall back on bad habits. The good news is that they actually know better, even if they don’t show it on a test. Also, there are things that can be done to work with students to help them address their stress during assessments – it just takes time, perseverance, patience, commitment, and buy-in.

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• Mark Pettyjohn

Hi Ethan, ‘cancelling’ and ‘undoing the multiplication’ may be a different issue but it appears to be the fundamental issue causing problems here. I don’t think it’s misunderstanding, more no understanding that cancelling happens because of multiplicative inverses (you could include a general lack of understanding of the properties of operations here as well). My perspective is elementary and middle school math so to me this appears that we have some improveming to do for you high school folks.

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• “I’d love to go back to pre-Algebra classes and remove canceling from our vocabulary. Change it to undoing the multiplication, since that’s really what we are doing. ”

Thank you: I’ll work to remember that when I next teach pre-Algebra.
Shira

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• Bruce Rhodewalt

One of the many problems with “cancel” is that it can be used to mean different things:
1. Subtract 1 from both sides of 2x + 1 = 17? Cancel the 1s!
2. Divide both sides of 2x = 16 by 2? Cancel the 2s!
3. Log base 3 both sides of 9 = 3^x? Cancel the 3s!

I think of “cancel” as a cancer. We keep trying to get each new occurrence, but it metastasizes.

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5. Twas simpler in the olden days,(that is the pre-calculator era), when logarithms first appeared as a means of easing “big sums”. The scientific calculator can help a lot here, with its e^x key, whose shifted operation is log base e. This should get the “it’s a function” idea across fairly painlessly.
I have a funny feeling that the log function is only in the syllabus in order to provide a method for doing yet more integrations in calculus !

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• I really disagree with this. It’s one of the best examples of the creation and use of inverse functions. If this is the focus, there is a lot that comes naturally, including the relationship between the laws of exponents and logarithms, and the means to solve problems that were previously inaccessible. I agree that if logs are taught just as “one more new thing to learn” this is bad, and it also needs to be taught as not just a means of easing a big multiplication or division.

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6. Introducing it as L(x) is a really strong idea — it’s those multi-lettered function names that really get them!
The troubling thing for me is the students who are able to say the function in English (I read “the number of factors of 2 it takes to build 8) and compute them on their own, yet still want to “cancel” when presented with a fraction. I wonder if it’s the same mechanism that leads students to remove a +3 in expressions like (2x + 3)/(4x + 3) — a “remove what’s the same” heuristic that simply overpowers their conception of log, or if it’s something else that’s unique to the multi-lettered functions.

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• Honestly i banned the word canceling from my math class. It becomes a reflex in a bunch of situations, and kids don’t distinguish combining opposite terms from dividing and making ones. These are different, and as soon as you stop to think about it, the slash marks to “cancel” the logs make no sense.

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• The blind application of the “cancellation rule” suggests to me that students do not understand that x is a number, and that any rule applied to an expression should work arithmetically for any particular value for x, and that this provides them with a checking method.

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• Of course, if they think that 6+3 divided by 12+3 is equal to 6 divided by 12 then the problem lies much further back.

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7. Ahhh, logarithms. Another favorite mistake has logA – logB = logA/logB because, well, there is a rule that turns subtraction into division, right? I certainly agree that the function notation business is a deep issue. However, a deeper issue here comes into play when students approach learning mathematics as a bunch of rules rather than a handful of principles. The undoing idea is a crucial one. I will often point out to my older students that the ‘dirty secret’ of algebra is that everything we learn to do we learn to undo resulting in a net progress of zero in a certain way. I also like the L(X) notation and it reminds me of my favorite algebra text that used INV(X) as the inverse notation rather than the -1 exponent notation. I think that my big takeaway here is to sit down with my Algebra II tam to discuss language use much more carefully before approaching our log unit. Thanks for the post and the good conversation in the comments here.

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• The concepts of named dependent and independent variable do rather cloud the simple concept of function as process, in which f(x) = x+ 3 and f(w) = w + 3 are definitions of the same function f. A simple diagram is quite useful:
input(x if you like)—–>f——–>output (y if you like)——>inverse of f—–>original input
and with the function bits in boxes it looks better.
I often wonder why students never seem to check anything.
I never gave my students the answers to problems. They complained. Compare with someone else I said. What if we get the same. then you are both right or both wrong!!!! Go figure.

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8. I always give the example to my students of sqrt(2) divided by sqrt(3) is NOT equal to 2/3 because sqrt is not a number or variable but a function as is the log.

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9. Matt

One thing you might say to a student who made that first error is this: “let’s say you have (5•6)/(3•4). Do you cancel out the multiplication signs to get 56/34? No, you would never do that, which is another reason why you can’t just cancel out the log functions.

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10. I really like the way you laid this out, Lani. I also like how you have students explain what a function is doing as well as having students explain about doing and undoing. I use the phrase “what you do to one side, you do to the other”, but doing and undoing sound much simpler. I am also going to reference this when teaching inverse functions this week and when we do logs and exponential functions, since the Tennessee Algebra II standards stress these topics. 🙂

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