So I have this idea.

My College Algebra students don’t get logarithms. I’ve got a bunch of ideas for improving the mathematical flow of our study of logarithms this semester, mostly having to do with lots of “find the exponent” tasks.

And I also want to provoke them to ask some questions of their own about logarithms. No small task, I assure you. Logarithms are abstract beasts.

So I’m thinking about where in the real world one encounters logarithms. The Richter scale and pH are my go-to examples. The Richter scale, while interesting (and of-the-moment), doesn’t suggest to me ways to bring it into the classroom to make things problematic for my students. I’m unsure how to get them wondering about the Richter scale.

So I’m going the pH route.

Here’s the setup: Orange juice concentrate. We have various mixtures of orange juice (which is acidic, so low pH). We will examine the pH of several of these mixtures and the question will be *What is the pH of the concentrate?*

So today I got some pH strips from our Chemistry department (props to Sara the CLA by the way-very helpful!) and bought some grapefruit juice in the College Store and mixed it up with water in various concentrations, testing each one with my strips.

Surprised by my results, I ran some computations.

So here’s the question for all you algebra teachers. Answer from your gut, not from computations. You have 10 seconds after reading the question to formulate the answer.

I can find it algebraically, and I know you can too, so no need to show off. Commit yourself to an answer.

(Readers who are not algebra teachers will find the answer in the comments section, but not a worked out solution.)

### The question

My grapefruit juice has pH=4 and my water has pH=7. I mix them in a 50-50 ratio. What is the pH of the resulting mixture?

Remember, answer from your gut. Then calculate.

### stay tuned

The task goes into full production tomorrow. Because of the surprising (to me, anyway) results of my grapefruit experiment, I now know for sure that pH strips are not good enough for my task. I need a pH meter and I’ll need to make special arrangements in the Chem Lab to make it happen.

**Update: **Special permission in chemistry obtained. Storyboard made. Filming scheduled for Thursday afternoon.

The new pH is 4.3

My gut has been informed form recently thinking about Dan Meyer’s coke vs. sprite thing. Because I think ph 7 means very little concentration (of hydrogen ions). So I’m a scientist, and I’m gonna say that’s like 0%. Then I’m just thinking, by ph 4 that’s like a very small fraction of a percent, but a whole lot more than 0%. In my mind, ph 4 dominates, and so you don’t get very far from ph of 4, but closer to 7.

This is great. Simple, accessible, and totally doable. Thanks! This kind of post is why blogging is so important!

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pH experiments are a good idea, but you could run into trouble with weak acids being buffered and not change the way you expect by diluting them.

Strong acids (say sulfuric) are likely to be a better choice, even if you use them in very low concentrations. Talk with a chemist.

My gut reaction is that a factor of two should be about 0.3 change in pH, but I didn’t do the computation.

I use logarithms all the time for representing probabilities in a computer. If you have to multiply together a lot of probabilities you get very small numbers, and on a computer that results in “floating-point underflow” and erroneously calling the small probability 0. If you use log probability in the computer, you end up adding instead of multiplying, and there is no danger of underflow.

Of course adding probabilities in log space is a bit messy: you end up needing to compute log(e^x+e^y), but that can be simplified to x+log(1+e^(y-x)), which can be handled in various ways, including simple table lookup of log(1+e^z) for z<0.

gasstation writes:

Hmmm… I have no idea what “buffering of acids” means. It seems, however, that one of the great things about a logarithmic scale is that even moderately-sized experimental errors get (seemingly) wiped out. In the video, we got pH for the water at 7.2 and for orange juice 3.8. The 1 ml concentrate in 100 ml water reading was 4.8. When we ran the numbers in class, we got 4.78. Close enough to convince ourselves of the correctness of the model.

What is the “factor of two” to which you refer? It seems to lead to an awfully nice guess, but why?

Interesting and clever use.

Have you much experience with College Algebra students at a community college? Recall that my goal was to find something

tangibleto bring to the classroom, and that logarithms have been particularlyintangiblefor them. If I were making a top-five list of mathematically intangible topics in this course, probability would definitely be among them.Perhaps you have something in mind that could help bring these ultra-small probabilities to life for this population. Or any probabilities, for that matter. I would certainly be quite grateful.

But for me one of the challenges of teaching probability is the very different nature of truth in the field. We could see that our gut instinct pH predictions were very, very wrong. The falsehood of our intuitive models was incontrovertible.

Probability is quite different. We run some probability computations and then we do an experiment. If we run it once, the thing either happens or it does not. If we run it many times, the proportion of successes

approximatesthe computed value. But we also do a lot of hand-waving about this approximation:Of course if we flip a coin 100 times, we don’t always get 50 heads, but it should be close…So for applications of logarithms, the small-probabilities example is a lovely one. But as a teaching tool for motivating logarithms in my College Algebra course, it doesn’t strike me as promising.

I agree that probability is not a good example for remedial algebra classes, as even students doing well in calculus often have terrible intuition about probability. Small probabilities come up a lot in information theory: the information content of a message in bits is -log_2(probability of message). This comes up all the time in data compression and in stochastic modeling, but framing that in away that would work in a remedial algebra class would be a major challenge. You could perhaps work it into a monkeys and typewriters problem: what is the probability of typing a page of text perfectly if random letters, spaces, and punctuation characters are typed? Assuming about 70 different characters, and about 1800 characters on a page, you get 70^-1800, which calculators will underflow to 0. If you do it in bits, each character is about 6.129 bits, and 1800 of them are about 11033 bits.

The 50-50 ratio is where I got the factor of 2. You were diluting the concentration of the citric acid by a factor of 2, and so the H+ concentration should drop by a factor of about 2. Since log_10(2) is about 0.3, that is the change I expected from the grapefruit juice.

Buffering of weak acids and bases comes from an equilibrium reaction: adding a bunch more H+ or OH- to a solution results in much of it being absorbed by the reaction, and the total change in pH is much less than you would expect from simple concentration calculations. http://en.wikipedia.org/wiki/Buffer_solution has a good explanation.