Logarithms: Some teaching ideas
Make them real for students
Ask a student about logarithms a year after they learned it in Algebra II, and you will usually hear a groan at the painful memory or “I never understood logs!”. PreCalculus teachers come to expect that knowledge of logs fades quickly and so they must review it before digging deeper.
I don’t have the magic cure for this (sorry!) but I do have a few real-world applications/demos that I’ve found useful in helping the concept at least feel more meaningful.
Decibel examples and experiments
Like many teachers, I use the decibel scale as a favorite example, accompanied by demonstrations.
Sensation of sound: I point out that, the energy from a sound source is inversely proportional to the square of the distance of the source to the listener. This is why earbuds sound impressive even though they put out just a tiny bit of sound energy – they are very close to your eardrums.
Standing say 16 feet from a student, I talk in a strong but normal-level voice, and ask how loud it sounds. She will typically say “not too loud”. I then say that I am going to increase the sound level so that it is 64 times as loud – I hope I don’t hurt her ears! I proceed to walk until I am about 2 feet away, while keeping my voice at the same level. I ask again, and usually the answer is “only a little louder”. Because our senses, in this case hearing, react in a logarithmic way, we need a logarithmic scale where, say, an increase of 100 times as loud feels like just a notch up. Indeed, the decibel scale works exactly this way, and this factor of 64 increase is only an increase of 18 decibels, perhaps from 60db to 78db which is about the way it feels. (This is an example of Fechner’s Law, something I hope to post about later.)
Producing sound: There are inexpensive or free smartphone apps that provide a decibel readout of the ambient sound. I put one on a projector display, and invite students to clap and see how high they can get it to read. They can see that it takes a substantial amount of applause to get the meter to move up very far.
Once the formula has been presented, I do another demo. I count-off 1,2,1,2,1,2,… around the room, dividing the students into two groups. I ask group #1 to applaud, and take a reading (without letting them see), I ask group #2 to applaud, and get a similar reading. I reveal the readings. I then ask for a prediction about what the reading will be if both groups applaud at the same time, doubling the sound energy. Based on the formula for decibels, the answer is that it should increase only 3.01 decibels (10 log 2), typically from about 88 to about 90 or 91.
As long as you are on the topic of decibels, you or your students can take some readings around the school. I’ve used a full cafeteria and the gym during a basketball game, but even readings around the house pique their curiosity.
History of the term “decibel”: A nice aside is to point out that the decibel is 1/10 of a “bel”, and the “bel” unit was named after Alexander Graham Bell (1847-1922) who is generally credited with the invention of the telephone.
At the end of the lesson one can tell colleagues that it was such a great lesson that students kept breaking into applause <grin>!
Semilog scales/graphs
In general, human sensation spans many, many orders of magnitude (powers of 10). For example, the ratio of the brightest light comfortable to our vision, to the dimmest light we can perceive, is about 10,000,000,000 (ten billion). Similarly, the ratio of a loud concert (120 db) to a whisper (10 db) is about 1,000,000,000,000 (1 trillion).
This presents a problem when we try to look at data quantitatively or visually. For example, here’s a bar chart showing sound energy values for some common situations:
It’s not very useful because typical sounds are so much lower than the loudest we can’t capture them all on one graph! Here’s the same chart in a log scale (db):
The defeat of Dracunculiasis (and a tribute to Jimmy Carter)
Whenever possible, I like to use the mathematical topics we are studying as an excuse to make students aware of things of importance. The past two years I’ve used the success, connected with the Carter Center and Jimmy Carter, in nearly eradicating Dracunculiasis (also known as Guinea worm disease). WHO has information on this (https://www.who.int/news-room/fact-sheets/detail/dracunculiasis-(guinea-worm-disease)) and also the Carter Center (https://www.cartercenter.org/health/guinea_worm/index.html)
Even though the annual cases have gone from nearly 1,000,000 and now hover between 10 and 15, we can display them on a single chart by using a log scale.
https://ourworldindata.org/grapher/number-of-reported-guinea-worm-dracunculiasis-cases
We can tell that the vertical axis is a log scale because each vertical increment is a factor of 10. If we take the log of each value, the scale become 2, 3, 4, 5, 6 as in a more typical scale.
This connection allows me to talk about the struggles of people living in poor conditions around the world, and to highlight the fact that it is possible to do something about it that can alleviate great suffering, while staying true to the curriculum and mathematics.
Pepper Hotness Scale
The Scoville Pepper scale (based on a 2015 MathFest presentation by Eric Landquist based on work with
Andrew Douventzidis, both at Kutztown University): When dining at a restaurant that serves spicy cuisine, it is often difficult to direct the chef to make the food at the desired level of hotness. People who eat spicy food regularly may think a dish is just medium hot, while those who rarely eat such food may find it intolerable. The word “medium” has different meanings to different people.
The heat in spicy food most often comes from plants in the pepper family. There is a scale, the Scoville scale, that is used to quantify the hotness of a pepper. However, the range of values makes the scale confusing to use. Here is a sign showing the Scoville values for a number of peppers (taken from Wikipedia):
You can see in the image that the Scoville rating for these peppers ranges from near 0 (bell peppers you might find in a garden salad) to habanero peppers at 500,000. Ghost peppers (not in the photo) reach 1,000,000 Scoville units. Pure capsaisin is 16,000,000. Prof. Landquist developed the assignment below based on this information.
(Full disclosure: I have not used this as an assignment in my classes, but only talked about the ideas.)
A bit of history of logarithms
It’s helpful for students to understand that terms and concepts in mathematics come from real people doing real thinking. The word “logarithm” comes from the Greek “logos” meaning “ratio” and the “arithmos” meaning number. So a logarithm is a “ratio number”. As a log increases by one unit, the value increases by a ratio. For log base 10, every unit increase corresponds to a factor of 10, so the ratio between numbers with a given log value and the value one greater is always 10.
Logs were originally invented for the purpose of simplifying arithmetic. Multiplication is time-consuming by hand, much harder than addition. Tables of logs were created and widely used to convert multiplication and exponentiation problems into addition problems. For example, to multiply 1645 by 4391, we look up the log of each in the table, then add the logs, then return to the table and find what number the sum corresponds to. That is, we use the fact that log(MN) = log(M) + log(N).
Here is a snippet of the first widely used tables, Henry Briggs 1617.
http://www.pmonta.com/tables/logarithmorum-chilias-prima/index.html
Note that log(10) = 1.00000000000000. We can also see that log(2) = 0.3010299566398. All these results were found by laborious hand arithmetic, of course! Once they were published, though, much labor was saved.
You can see in the table the relationship between log(2) and log(4) and log(8)! What should the log(16) be?