Diagrams, week 7

We are studying logarithms in College Algebra.

We began with fact families. Many reps; some started with the exponential fact, some with the log fact, some with the root fact.

We expressed our heights as logarithms, base 10, of our heights in inches, and we brought a tall and a short student to the front of the classroom to stand next to each other. Given the log height of the tall student, we predicted the log height of the short one and were surprised to find that the difference was less than one-tenth of a log-unit (whatever that is).

We came to terms with this difference by noticing that the tall student, while much taller than the short student, was not exponentially taller. The tall student wasn’t even twice as tall-never mind ten times as tall.

I gave the definition of pH as the negative log of the hydrogen ion concentration. We considered the consequences of this goofy definition, and what it says about possible values on the pH scale. We puzzled over why pH stops at 14, when there should be no theoretical limit on the upper value of a negative logarithm of a positive ratio, and we wondered why 7 is considered neutral.

Then we watched the video in which I measure the pH of orange juice concentrate and of water (with an imperfectly calibrated pH meter-so don’t give me grief about the precise values involved here). I reminded students of our height-measuring surprise and summarized our knowledge of the acidity of water and of orange juice with the diagram below:

Finally, I asked them to predict the pH of a 50/50 mixture of water and orange juice concentrate.

Nearly all of them averaged the two pH values (of course), getting 5.7 or some adjusted value nearby.

We watched that video and saw that the result is nearly indistinguishable from the original pH of the orange juice concentrate.

We were surprised.

But someone noticed that the difference of roughly 3 in the pH values means that the hydrogen ion concentration is 10,000 times as much in the orange juice concentrate as it is in the water.

Just like our tall and short students had a big difference in height, but small difference in log heights, our two substances must have an enormous difference in hydrogen ion concentration to account for the sizable difference in pH values.

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Can’t get enough of logs!

You know what fact families are? In elementary and middle school, this is increasingly common terminology to express relationships among basic number facts. A sample fact family:

 

Fact family thinking is intended to encourage connections between operations and their inverses. Addition and subtraction fact families work the same way as multiplication and division fact families:

 

I have written a lot about logarithms this past year and I keep wondering about ways to get students’ minds wrapped around the relationships that logarithms are intended to express, and to focus less on the notation.

Here is a fact family for an exponential relationship.

Some things I notice:

1. We’re not used to thinking about roots, logs and exponentials all at the same time. Maybe we should be.
2. Exponentiation is not commutative, so we only have three facts in the family, not four. That seems useful to know.
3. Off the top of my head, I can’t think of another non-commutative binary operation (besides those listed out above), so I can’t test the proposition that:
4. All non-commutative binary operations will necessarily have TWO inverse operations in their fact families.

Half lives

Man do I hate this table:

If you get the basic principles at play here, the table is needless. And if you don’t get the basic principles at play, you would be much better served studying those principles than studying this table. (How do I go from yearly to daily, for instance? That seems to have been left out of the table.)

And at the College Algebra level, I hate this equation:

.

That’s because there is no good explanation of e that is justifiable with the tools of College Algebra. You want to describe it as a limiting value of interest rates as number of compounding cycles increases? You’re gonna do a lot of hand-waving at the end there. (Come to think of it, that’s not a bad motivating question for limits in Calculus.)

And any other way you want to introduce e (and mathematicians’ fascination with it) is going to rely even more heavily on Calculus.

But half-lives are a nice, tidy application of an important set of College Algebra topics: exponential and logarithmic relationships.

So I give an extended assignment that deals with half-lives. And I encourage students to think in terms of an annual decay factor, which we can find by means of extracting roots. We know that it takes thirty years before the decay factor gets down to 1/2, so we need to know what number to the 30th power is equal to 1/2.

Basically, I hope to get them to the point where they want to solve this equation:

And in reading their work, I have been blown away by the number of students who have done the whole assignment in terms of half-lives. They are thinking explicitly in terms of half-lives. Rather than blindly plug numbers into , they are figuring out how many half-lives there are and then using that number to calculate. So they’re using the equation, , where x represents the number of half-lives that have passed.

They are putting appropriate scales on their axes, labeling the x-axis half lives. And one even made the observation that, with this setup, the graph describes all half-life situations; the only difference is the length of the half-life.

I don’t hate that.

What if we spoke about division as we do logarithms?

Definition: The quotient, factor a, of c is b if and only if ab is c. We introduce the following notation:

Examples

When we read this notation aloud, we say (for the second example), The quotient, factor 5, of 10 is 2.

The important thing to remember about a quotient is that a quotient is a factor. When we see this equation:

we ask ourselves, What factor goes with 3 to make 12?

Properties of the quotient

The following properties of quotient are derived from corresponding properties of multiplication.

laws of the quotient

There are some important laws pertaining to the quotient. The next section will include algebraic proofs of these laws.

These properties will be important when solving equations involving quotients.

factors for the quotient function

Standard quotient tables and calculator functions involve one of two factors: 10 or u, where u, the factor of the natural quotient, is defined by the series below:

Quotients using this factor can be notated one of two ways:

Quotients factor 10 are so common that they also have special notation, so that the following two equations are equivalent in meaning:

One other convention for the factor 10 quotient is this: If a factor is not indicated, we assume the quotient is factor 10. More commonly, we capitalize the Q in Quot in order to indicate the factor 10 quotient but either notation is accepted.

Many times, we will be able to find a quotient by inspection and use of the associated known multiplication facts. In almost all cases involving relatively small numbers, we can approximate the value of a quotient by use of multiplication facts.

When we require a greater level of precision, we will need the change of factor formula.

Change of factor formula

Applications of the quotient

The quotient function appears so often in computations that some important measures are defined in terms of it. Of these, perhaps the most important is speed.

The speed (r) of an object is defined in terms of the distance (d) it travels over the time period (t) in following way:

A kernel of an idea…(logs)

The standard explanation is this, “a logarithm is an exponent”.

This is true. But I’m not sure it’s particularly helpful for a student who is struggling. I have been burning a lot of mental calories over the past few versions of College Algebra trying to come up with ways into logarithms that will have more explanatory power and be more intuitively inviting to my students.

I realize the quest may be quixotic.

And I understand that this is well-trodden ground.

But consider this equation:

In the standard interpretation, this asks, What exponent do I put on 2 to get 9?

What if instead, we thought about it this way: How many factors of 2 are in 9?

More than 3, fewer than 4.

One has no factors of 2 in it, so:

But even better,

because the expression on the left asks how many factors of 2 are in AB while the one on the right asks how many are in A and how many are in B, then adds them. Some of the 2s are in A, the rest are in B.

The language is problematic, I know. The answer to the question How many factors of 2 are there? is properly “2”.

But we use the language of factor in our exponential work, so it may not be too problematic in this context (and few College Algebra students are number theorists).

And of course I’m not naive enough to expect that this will solve all of our logarithm difficulties. But I’ve got something to work with.

Thoughts and critiques?