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.
Lovely! I’ve never focused much on the log scales. (More on using logs to solve exponential equations.) But you are convincing me I can help students to understand these ideas. I’ll include this in my next unit in Pre-Calc. Thanks!
Thanks, Sue. I’m glad this progression seems useful to others as well. I find that the older I get, the less patience I have for disconnected skill. I end up crafting story lines such as this one that build up to the skill. If you forget the skill (is log of 0 equal to 1 or is log of 1 equal to zero?), you can get it back by telling yourself the story.
Where we’ll go from here is to logarithm as a function. We’ll ask what characteristics of the graph will explain the height conundrum, and what characteristics will explain the pH one. It’s a ton of fun.
And I have to say that it is astonishing to have a group of College Algebra students arguing intelligently-and on the basis of mathematical principles-about why the pH of orange juice is nearly indistinguishable from the pH of orange juice concentrate.