# Sliderule for teaching logarithms

I recently learned how to use a slide rule and I have since been wondering why they are not commonly used to teach properties of logarithms. It seems that they are very clear examples of the $\log(a) + \log(b) = \log(ab)$ rule and several other similar ones. This seems like a concept that many students struggle with, and the slide rule seems to provide both a visual and possibly even kinetic representation of it. I have not yet seen an application of slide rules in the classroom despite the obvious usability for logarithms. Is there a reason that they are used so uncommonly? (or am I simply oblivious to the slideruling masses?)

• This is just a comment -- I dig this question, because the reason I first got interested in math has to do with all those times I snuck my dad's old slide rule from his desk and messed around with it, and logarithms/exponentials left a vivid impression on me in particular, years before I learned what those things were. I personally would love to see more of a physical component in computation when learning basic math. – Robert Talbert Mar 3 '16 at 22:42

Anecdote: For some years I've been trying to get my hands on a slide rule and I can't find any vendor who sells them.

This highlights the fact that for general purposes the electronic calculator has made the slide rule defunct, to the extent that I don't think anyone even manufactures them in scale any more. So making them available in the classroom for a fairly limited exercise would be difficult.

Or to turn the issue around, the biggest reason historically that logarithms were taught in the first place was to support calculations using the slide rule (or equivalent tables). Granted that calculators have superseded slide rules, logarithms aren't even taught that widely anymore (I got them in high school in the late 1980's, but that was presented as an "optional advanced" topic then.)

However, there are web/smartphone applications that simulate functioning slide rules, so you might consider looking for those instead.

• ebay.com/itm/… – JTP - Apologise to Monica Feb 16 '16 at 3:46
• That looks very doable for a single sliderule, but if you wanted to get them for an entire class, it would get very expensive very quickly – nosyarg Feb 16 '16 at 4:07
• Here's a thread from 2012 that someone responded to just last week: physicsforums.com/threads/… – shoover Feb 16 '16 at 17:48
• My slide rule was a free gift from my secondary school. They had lots sitting in a store cupboard which hadn't been used in years. It might be worth asking the older teachers at the next conference you attend whether they have a set of 20 or more gathering dust. – Peter Taylor Apr 16 '19 at 7:33

As Daniel's answer states, the most likely reason why slide rules are rarely used to teach logarithms boils down to them being hard to come by. As they have been replaced by electronic calculators, it is common for many educators now to have never seen one, or to only be peripherally aware of them. Many don't even think of including them in lesson plans.

However, this doesn't mean you shouldn't. In fact, if you have the time available, and you don't have a slide rule to hand, you can do something even better - have your class make them!

This doesn't take long (<20mins), and I personally find it a demonstration of laws of logarithms that is so obvious and intuitive that it seems almost "too obvious". You will need the following:

1. Standard ruler
2. Pen/pencil
3. 2 strips of card/paper (just cut a standard piece length-ways down the middle)

Now align both pieces, and make a mark along the edge every 2 cm (other graduations are available). Write a simple linear scale on the top and bottom piece such that each gradation is 1 added to the value of the previous gradation.

Now, use your simple scale to perform an addition. If you want perform $$2 + 3$$, offset the top scale so the zero lines up with the $$2$$ on the bottom scale. Now find the $$3$$ on the top scale, and look at the number below it on the bottom scale. Tada! 5!

Ok, at this point, who cares. We are only introducing the idea that two parallel sliding scales can add two values together. Now lets make a real slide rule.

Flip your pieces of card over, and make a new set of graduated scale lines. However, instead of incrementing by one each time, multiply by some base. I use two. It is worth pointing out that this is a logarithmic scale and each entry is just the linear scale value, raised as a power of two.

I think it helps to make a larger mark for the $$0$$ line, and to write $$2^n$$ above each graduation.

Now perform the same operation as before, only as a multiplication. We will find $$4 \times 8$$. Line up the $$1$$ of the top scale with the $$4$$ on the bottom. Find the $$8$$ on the top scale, and then look at what it lines up with on the bottom. $$4 \times 8=32$$

Congratulations, you have now made a simple slide rule, and used it to demonstrate the first law of logarithms. The action that we performed is the same "addition" action we performed with the linear scales, but instead of adding, we ended up with a multiplication, because our scales are logarithmic.

If you have more time, you can use this to explore further concepts:

• 2nd and 3rd laws of logarithms
• Different bases (use a 10 times multiplier instead of 2)
• Increase the precision. Use a calculator to find the logarithmic values of numbers in-between the ones already present, and mark them in the correct place on the scale. This will allow you to perform multiplication with more values, and helps familiarise students with logarithmic scales.
• I also do this! I would recommend labeling the picture differently -- label the "4" in your last image as "$log_2 4$" and the "8" as "$log_2 8$". Then the overall distance is "$log_2 32$"! – Chris Cunningham Apr 17 '19 at 16:28
• Absolutely brilliant - I've long believed that logs are best introduced by starting with base 2 examples, and this only reinforces that idea. Also, if the project is too time consuming for students to each construct, not too tough to do this once and make photocopies. – JTP - Apologise to Monica Apr 17 '19 at 21:36
• As a first step to fill in numbers other than powers of 2, you can benefit from the fact that $2^{10}$ is very close to $10^3$. So if your slide rule is long enough to have a marking for 1024, just trisect the segment between that mark and 1, and you have a pretty accurate location to put 10. Then you can do 5, 20, etc. by multiplying by powers of 2. ("Pretty accurate" here means "no worse than the inevitable sloppiness of dong the whole construction by hand.") – Andreas Blass Apr 18 '19 at 22:00

Use printed log tables instead. You see the patterns more clearly than with a calculator or a slide rule. We learned how to use them in chemistry class--they were still provided on the AP chem exam until the late 90s.

In addition, instead of needing to remember, prove or the like the addition of logs, you are DIRECTLY USING it, repeatedly. Nothing motivates belief and memory like actually having dependence on the property. Have a choice between manually adding logs from a table or doing the multiplication of the original numbers and it is a no-brainer which you go to. You also get very used to characteristic and mantissa.

I never used the log tables much after the 70s (calculators became cheap). But just that initial use in high school chem was very powerful for me. Much less cryptic than a button on a box or moving a ruler stick side by side. (It also set me up to use trig and steam tables and later Bessel function tables, probability tables, financial tables, etc. Just developing a facility with tables is good thing to have. You still need it.)

See:

https://www.youtube.com/watch?v=VRzH4xB0GdM (more motivating and slick but not as quick to just show the tables)

or

https://www.youtube.com/watch?v=4ebnr_h4m7E (very hands on and practical, but heavily accented English)

To print tables for your class just zox the few pages out of your CRC. (Of course you have a CRC. No math/science person would not. Plus it is good for ganglion bashing.)

P.s. Please start with common logs, not Napierian. As a veteran scientist or mathematician, it is fine to use ln instead of log because we already know that stuff. But for kids just learning the thing for first time, base 10 is much easier. You see the exponents of 10, 100, 1000 etc. very clearly.

• Regarding the use of logarithms (see here for how we did it when I was in high school), your comments reminded me of similar things I've said (and even discussed in a $T^3$ conference talk in November 1998) about scaffolding --- see post 1 post 2 post 3. – Dave L Renfro Apr 17 '19 at 8:36
• Thanks, Dave. You're the man. P.s. slight segue but related to theme of your talk: salon.com/2006/09/14/basic_2 – guest Apr 17 '19 at 9:45