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I found that looking at the original motivation of logarithms has been more elucidating than the way the topic is presented in grade-school. Thinking through th

I found that looking at the original motivation of logarithms has been more elucidating than the way the topic is presented in grade-school. Thinking through the functional form that can solve the multiplication problem that Napier was facing (how to simplify multiplying large astronomical observations), f(ab) = f(a) + f(b), and why that leads to a unique family of functions, resonates a lot better with me for why logarithms show up everywhere. This is in contrast to teaching them as the inverse of the exponential function, which was not how the concept was discussed until Euler. In fact, I think learning about mathematics in this way is more fun — what original problem was the author trying to solve, and what tools were available to them at the time?

cauliflower2718 2y ago
This follows directly from the fact that exp(x+y)=exp(x)exp(y).
- dkislyuk 2y ago
  Yes, but such a property was not available to Napier, and from a teaching perspective, it requires understanding exponentials and their characterizations first. Starting from the original problem of how to simplify large multiplications seems like a more grounded way to introduce the concept.
  - kccqzy 2y ago
    From a teaching perspective it goes like this: first we learn additions, and to undo additions we have subtractions; then we learn repeated additions i.e. multiplications, and to undo multiplications we have divisions; finally we learn repeated multiplications, i.e. exponentiation, and to undo exponentiation we have logarithms and roots.
    - BobaFloutist 2y ago
      You see how one of those isn't like the others?
      - kccqzy 2y ago
        You mean we have both logarithms and roots to undo exponentiation? That's because exponentiation is non-commutative.
        BobaFloutist 2y ago
        Right, I'm not saying it's for no reason, but the asymmetry makes it harder to keep track of which undoes exponentiation in which way. And logs are frankly more confusing than the other operations because more than anything else they feel like an algebraic expression in the form of an operation. Other operations intuitively feel like a process, whereas logs feel like more like a question. Maybe that's just because I never learned them super well though, maybe they're not actually that inherently different ¯\_(ツ)_/¯
saulpw 2y ago
I think this should be front and center. To that end I propose "magnitude notation"[0] (and I don't think we should use the word logarithm, which sounds like advanced math and turns people away from the basic concept, which does make math easier and more fun). https://saul.pw/mag https://saul.pw/mag
- JadeNB 2y ago
  > I think this should be front and center. To that end I propose "magnitude notation"[0] (and I don't think we should use the word logarithm, which sounds like advanced math and turns people away from the basic concept, which does make math easier and more fun). The only reason that "logarithm" sounds like advanced math is because it was so useful that mathematicians, well, used it. Since this terminology is just logarithms without saying the word, if it is more useful it, too, will probably be used by mathematicians, and then it will similarly come to sound like advanced math. So what's the point of running away from a name for what we're doing that fits with what it's actually called, if eventually we'll just have to make up a new, even less threatening name for it? (I'd argue that "logarithm" is frightening less because it sounds like advanced math than because it's an unfamiliar and old-fashioned-sounding word. I'm not completely sure that "magnitude" avoids both these issues, but it's at least arguable that it suffers less from them.)
  - saulpw 2y ago
    It's written like ^6 and said like "mag 6", which sounds like an earthquake (and this is basically the Richter scale writ large). One syllable, sounds cool, easy to type/spell, evokes largeness. "Logarithm" is 3-4 syllables, hard to pronounce, hard to spell, sounds jargon-y.
    - seanhunter 2y ago
      People virtually never say “logarithm” in use though. They either say “log” or they say “lun” for natural log. Notice that both log and lun are one syllable, easy to pronounce etc. Magnitude is an existing and important concept in maths - it would be extremely confusing to just overload it to mean something else.
agumonkey 2y ago
I often wonder about this. I also believe that mathematical pedagogy strive to attract people that are very smart and think in the abstract like euler, and not operationally, meaning they will get it intuitively. For other people, you need to swim in the original problem for a while to see the light.
- BobbyTables2 2y ago
  Math is rarely taught with practical problems in mind — that’s engineering !
  - smitty1e 2y ago
    Therein lies the rub. Treating abstract and the concrete in isolation was always tough sledding for me. Bouncing between the two is where the action is. And units: if I had it all to do over, I would pore over the units sooner rather than later.
    - pbronez 2y ago
      Absolutely. Units are such a useful idea. I was recently struggling to model a financial process and solved it with Units. Once I started talking about colors of money as units, it became much easier to reason about which operations were valid.
      - kqr 2y ago
        Strictly speaking this is about dimensional analysis, not units. (When discussing curricula we should be precise!)
  - fenomas 2y ago
    Agreed, and it's such a shame! A kid goes to math class and learns, say, derivatives as this weird set of transformations that have to be memorized, and it's only later in in physics class that they start to see why the transformations are useful. I mean, imagine a programming course where students spend the whole first year studying OpenGL, and then in the second year they learn that those APIs they've been memorizing can be used to draw pictures :D
malshe 2y ago
We used logarithms routinely for large multiplications, divisions, etc. in 11th and 12th grade. No calculators were allowed. This was in India.
- ninalanyon 2y ago
  Same here when I was at school in the late 1960s and early 1970s. No one had a calculator. So we were taught logarithms as a tool first.
meta_ai_x 2y ago
I actually prefer the straightforward log is an inverse of exponents. It's more intuitive that way because I automatically can understand 10^2 * 10^3 = 10^5. Hence if you are using log tables, addition makes sense. I didn't need an essay to explain that. Take logs, add 2 + 3 = 5 and then raise it back to get 10^5.
- kqr 2y ago
  This is how I've always taught logarithms to students I've tutored. I photocopy a table of various powers of ten, we use it in all sorts of ways to solve problems, and then I sneakily present an "inverse power" problem where they need to make the lookup backwards. Almost every student gets it right away, and then I tell them looking up things backwards in the power table is called taking a logarithm.
- Dunan 2y ago
  That's how I mentally processed them when first learning them years ago. Doing operations on x and y with log(x) = y in the background somehow felt far less intuitive than thinking about 10^y = x. I really enjoyed this author's work, BTW. Just spent several hours reading the entire first five chapters or so. What an excellent refresher for high school math in general.
analog31 2y ago
This would be an interesting thing to study: How many different ways people learned about logarithms, and how they generally fared in math. I learned about logarithms by seeing my dad use his slide rule, and studying stock charts, which tended to be semi-logarithmic.
madars 2y ago
Toeplitz wrote "Calculus: The Genetic Approach" and his approach of explaining math via its historical development is apparently more widely used: https://en.wikipedia.org/wiki/Genetic_method https://en.wikipedia.org/wiki/Genetic_method . Felix Klein remarked: "on a small scale, a learner naturally and always has to repeat the same developments that the sciences went through on a large scale"
- b0afc375b5 2y ago
  I always longed for a book/course on mathematics where topics are in chronological order: 1. ... (mathematical topics at the beginning of history of which I am ignorant of) 2. pythagoras theorem 3. ... 4. euclid geometry 5. ... 6. algebra 7. ... 8. calculus 9. ... 10. set theory 11. ... 12. number theory 13. etc. etc. (you get the point) Maybe there's already something that lays out topics like this. I haven't searched too hard.
  - twelvechairs 2y ago
    I'm sympathetic but there's no clear historic chronology. For instance the ancient egyptians dealt with both algebra and calculus (at least in part) long before Pythagoras. And thats not starting on China and India which had very different chronologies.
    - PaulRobinson 2y ago
      Choose a chronology that makes sense. We can see how Western ideas build, we have less clarity on how the ancient Egyptians or Chinese ideas developed, and therefore it's harder to explain to a learner. If you're sensitive to that singular world view warping the learner's prospect, you could at each point explain similar ideas from other cultures that pre-date that chronology. For example, once you've introduced calculus and helped a student understand it, you can then jump back and point out that ancient Egyptians seemed to have a take on it, explain it, ask the student to reason did they get there in the same way as the Western school of ideas did, is there an interesting insight to that way of thinking about the World? Another ideas is how ideas evolved. We know Newton and Leibniz couldn't have had access to direct Egyptian sources (hieroglyphs were a lost language in their life times), but Greek ideas would have been rolling around in their heads.
coffeemug 2y ago
Where did you pick this up? Is there a book that covers it that way?
- II2II 2y ago
  In my case, it was by chance. I had a slide rule in high school. It was more of a novelty item by that point in time, only one of my math teachers even knew what a slide rule was, but that didn't stop me from figuring out how it was used and how it works. It didn't take much to figure out that the sliding action was solving problems by addition, and the funky scales were logarithmic. In other words: it performed multiplication by adding logs. That said, I did encounter references to its original applications in other places. I studied astronomy and had an interest in the history of computation.
- dkislyuk 2y ago
  Presumably the book from this thread by Charles Petzold will be a great canonical resource, but originally there was a quote by Howard Eves that I came across that got me curious: > One of the anomalies in the history of mathematics is the fact that logarithms were discovered before exponents were in use. One can treat the discovery of logarithms as the search for a computation tool to turn multiplication (which was difficult in the 17th century) into addition. There were previous approaches for simplifying multiplication dating back to antiquity (quarter square multiplication, prosthaphaeresis), and A Brief History of Logarithms by R. C. Pierce covers this, where it’s framed as establishing correspondences between geometric and and arithmetic sequences. Playing around with functions that could possibly fit the functional equation f(ab) = f(a) + f(b) is a good, if manual, way to convince oneself that such functions do exist and that this is the defining characteristic of the logarithm (and not just a convenient property). For example, log probability is central to information theory and thus many ML topics, and the fundamental reason is because Claude Shannon wanted a transformation on top of probability (self-information) that would turn the probability of multiple events into an addition — the aforementioned "f" is the transformation that fits this additive property (and a few others), hence log() everywhere. Interestingly, the logarithm “algorithm” was considered quite groundbreaking at the time; Johannes Kepler, a primary beneficiary of the breakthrough, dedicated one of his books to Napier. R. C. Pierce wrote: > Indeed, it has been postulated that logarithms literally lengthened the life spans of astronomers, who had formerly been sorely bent and often broken early by the masses of calculations their art required.
JackFr 2y ago
This is how I learned them in middle school — just common logs, as an aid to doing roots, powers and multiplications of big numbers. We were told in an off-hand way that logs could be to any base, even ‘e’, but not to worry about that for a few years.
meindnoch 2y ago
By the way, there's another function that can be used to turn multiplication into addition: f(x) = x^2 / 2 a * b = f(a + b) - (f(a) + f(b))
- secondcoming 2y ago
  Isn’t x^2 a multiplication?
  - meindnoch 2y ago
    No, you misunderstood what I meant. Normally, a sliderule at distance x has the value of log(x) written on it, which allows doing multiplications by moving along the sliderule, since log(ab) = log(a) + log(b). Now imagine a sliderule onto which values of x^2/2 are written. This also allows you to multiply two numbers, because ab = (a+b)^2/2 - (a^2/2 + b^2/2).
pipes 2y ago
Coincidentally I watched this last night https://m.youtube.com/watch?v=7TWKSMtKCmU https://m.youtube.com/watch?v=7TWKSMtKCmU It gives the history / motivation behind logarithms and suddenly it became so much clearer to me. Pretty much multipling huge numbers by adding exponents , well I think I've understood that correctly? I think why I'm so interested in programming and computing is because I fascinated by the history of it all. It somehow acts as a motivation to understand it.
tim333 2y ago
I rather like Feynman's approach in the lecture Algebra from the Feynman Lectures https://www.feynmanlectures.caltech.edu/I_22.html https://www.feynmanlectures.caltech.edu/I_22.html He covers the inverse of the exponential, Henry Briggs' log tables and goes on to e^ix = cos x + i sin x The audio is also available https://www.feynmanlectures.caltech.edu/flptapes.html https://www.feynmanlectures.caltech.edu/flptapes.html
xiande04 2y ago
If you like this approach, I highly recommend Mathematics: It's Content, Methods, and Meaning by Kolmogorov. He uses this same approach, but applies it to many more concepts in math (about 1,000 pages!). In fact, I think I actually heard about that book on this site, so I guess I'm paying it forward. This approach was to align with the Soviet philosophy of dialectical materialism, which claims that all things arise from a material need. Not sure I'm fully onboard with the philosophy as a whole, but Kolmogorov's book was really eye opening.

cauliflower2718 2y ago

This follows directly from the fact that exp(x+y)=exp(x)exp(y).

dkislyuk 2y ago
Yes, but such a property was not available to Napier, and from a teaching perspective, it requires understanding exponentials and their characterizations first. Starting from the original problem of how to simplify large multiplications seems like a more grounded way to introduce the concept.
- kccqzy 2y ago
  From a teaching perspective it goes like this: first we learn additions, and to undo additions we have subtractions; then we learn repeated additions i.e. multiplications, and to undo multiplications we have divisions; finally we learn repeated multiplications, i.e. exponentiation, and to undo exponentiation we have logarithms and roots.
  - BobaFloutist 2y ago
    You see how one of those isn't like the others?
    - kccqzy 2y ago
      You mean we have both logarithms and roots to undo exponentiation? That's because exponentiation is non-commutative.
      - BobaFloutist 2y ago
        Right, I'm not saying it's for no reason, but the asymmetry makes it harder to keep track of which undoes exponentiation in which way. And logs are frankly more confusing than the other operations because more than anything else they feel like an algebraic expression in the form of an operation. Other operations intuitively feel like a process, whereas logs feel like more like a question. Maybe that's just because I never learned them super well though, maybe they're not actually that inherently different ¯\_(ツ)_/¯

saulpw 2y ago

I think this should be front and center. To that end I propose "magnitude notation"[0] (and I don't think we should use the word logarithm, which sounds like advanced math and turns people away from the basic concept, which does make math easier and more fun). https://saul.pw/mag https://saul.pw/mag

JadeNB 2y ago
> I think this should be front and center. To that end I propose "magnitude notation"[0] (and I don't think we should use the word logarithm, which sounds like advanced math and turns people away from the basic concept, which does make math easier and more fun). The only reason that "logarithm" sounds like advanced math is because it was so useful that mathematicians, well, used it. Since this terminology is just logarithms without saying the word, if it is more useful it, too, will probably be used by mathematicians, and then it will similarly come to sound like advanced math. So what's the point of running away from a name for what we're doing that fits with what it's actually called, if eventually we'll just have to make up a new, even less threatening name for it? (I'd argue that "logarithm" is frightening less because it sounds like advanced math than because it's an unfamiliar and old-fashioned-sounding word. I'm not completely sure that "magnitude" avoids both these issues, but it's at least arguable that it suffers less from them.)
- saulpw 2y ago
  It's written like ^6 and said like "mag 6", which sounds like an earthquake (and this is basically the Richter scale writ large). One syllable, sounds cool, easy to type/spell, evokes largeness. "Logarithm" is 3-4 syllables, hard to pronounce, hard to spell, sounds jargon-y.
  - seanhunter 2y ago
    People virtually never say “logarithm” in use though. They either say “log” or they say “lun” for natural log. Notice that both log and lun are one syllable, easy to pronounce etc. Magnitude is an existing and important concept in maths - it would be extremely confusing to just overload it to mean something else.

agumonkey 2y ago

I often wonder about this. I also believe that mathematical pedagogy strive to attract people that are very smart and think in the abstract like euler, and not operationally, meaning they will get it intuitively. For other people, you need to swim in the original problem for a while to see the light.

BobbyTables2 2y ago
Math is rarely taught with practical problems in mind — that’s engineering !
- smitty1e 2y ago
  Therein lies the rub. Treating abstract and the concrete in isolation was always tough sledding for me. Bouncing between the two is where the action is. And units: if I had it all to do over, I would pore over the units sooner rather than later.
  - pbronez 2y ago
    Absolutely. Units are such a useful idea. I was recently struggling to model a financial process and solved it with Units. Once I started talking about colors of money as units, it became much easier to reason about which operations were valid.
    - kqr 2y ago
      Strictly speaking this is about dimensional analysis, not units. (When discussing curricula we should be precise!)
- fenomas 2y ago
  Agreed, and it's such a shame! A kid goes to math class and learns, say, derivatives as this weird set of transformations that have to be memorized, and it's only later in in physics class that they start to see why the transformations are useful. I mean, imagine a programming course where students spend the whole first year studying OpenGL, and then in the second year they learn that those APIs they've been memorizing can be used to draw pictures :D

malshe 2y ago

We used logarithms routinely for large multiplications, divisions, etc. in 11th and 12th grade. No calculators were allowed. This was in India.

ninalanyon 2y ago
Same here when I was at school in the late 1960s and early 1970s. No one had a calculator. So we were taught logarithms as a tool first.

meta_ai_x 2y ago

I actually prefer the straightforward log is an inverse of exponents. It's more intuitive that way because I automatically can understand 10^2 * 10^3 = 10^5. Hence if you are using log tables, addition makes sense. I didn't need an essay to explain that. Take logs, add 2 + 3 = 5 and then raise it back to get 10^5.

kqr 2y ago
This is how I've always taught logarithms to students I've tutored. I photocopy a table of various powers of ten, we use it in all sorts of ways to solve problems, and then I sneakily present an "inverse power" problem where they need to make the lookup backwards. Almost every student gets it right away, and then I tell them looking up things backwards in the power table is called taking a logarithm.
Dunan 2y ago
That's how I mentally processed them when first learning them years ago. Doing operations on x and y with log(x) = y in the background somehow felt far less intuitive than thinking about 10^y = x. I really enjoyed this author's work, BTW. Just spent several hours reading the entire first five chapters or so. What an excellent refresher for high school math in general.

analog31 2y ago

This would be an interesting thing to study: How many different ways people learned about logarithms, and how they generally fared in math. I learned about logarithms by seeing my dad use his slide rule, and studying stock charts, which tended to be semi-logarithmic.

madars 2y ago

Toeplitz wrote "Calculus: The Genetic Approach" and his approach of explaining math via its historical development is apparently more widely used: https://en.wikipedia.org/wiki/Genetic_method https://en.wikipedia.org/wiki/Genetic_method . Felix Klein remarked: "on a small scale, a learner naturally and always has to repeat the same developments that the sciences went through on a large scale"

b0afc375b5 2y ago
I always longed for a book/course on mathematics where topics are in chronological order: 1. ... (mathematical topics at the beginning of history of which I am ignorant of) 2. pythagoras theorem 3. ... 4. euclid geometry 5. ... 6. algebra 7. ... 8. calculus 9. ... 10. set theory 11. ... 12. number theory 13. etc. etc. (you get the point) Maybe there's already something that lays out topics like this. I haven't searched too hard.
- twelvechairs 2y ago
  I'm sympathetic but there's no clear historic chronology. For instance the ancient egyptians dealt with both algebra and calculus (at least in part) long before Pythagoras. And thats not starting on China and India which had very different chronologies.
  - PaulRobinson 2y ago
    Choose a chronology that makes sense. We can see how Western ideas build, we have less clarity on how the ancient Egyptians or Chinese ideas developed, and therefore it's harder to explain to a learner. If you're sensitive to that singular world view warping the learner's prospect, you could at each point explain similar ideas from other cultures that pre-date that chronology. For example, once you've introduced calculus and helped a student understand it, you can then jump back and point out that ancient Egyptians seemed to have a take on it, explain it, ask the student to reason did they get there in the same way as the Western school of ideas did, is there an interesting insight to that way of thinking about the World? Another ideas is how ideas evolved. We know Newton and Leibniz couldn't have had access to direct Egyptian sources (hieroglyphs were a lost language in their life times), but Greek ideas would have been rolling around in their heads.

coffeemug 2y ago

Where did you pick this up? Is there a book that covers it that way?

II2II 2y ago
In my case, it was by chance. I had a slide rule in high school. It was more of a novelty item by that point in time, only one of my math teachers even knew what a slide rule was, but that didn't stop me from figuring out how it was used and how it works. It didn't take much to figure out that the sliding action was solving problems by addition, and the funky scales were logarithmic. In other words: it performed multiplication by adding logs. That said, I did encounter references to its original applications in other places. I studied astronomy and had an interest in the history of computation.
dkislyuk 2y ago
Presumably the book from this thread by Charles Petzold will be a great canonical resource, but originally there was a quote by Howard Eves that I came across that got me curious: > One of the anomalies in the history of mathematics is the fact that logarithms were discovered before exponents were in use. One can treat the discovery of logarithms as the search for a computation tool to turn multiplication (which was difficult in the 17th century) into addition. There were previous approaches for simplifying multiplication dating back to antiquity (quarter square multiplication, prosthaphaeresis), and A Brief History of Logarithms by R. C. Pierce covers this, where it’s framed as establishing correspondences between geometric and and arithmetic sequences. Playing around with functions that could possibly fit the functional equation f(ab) = f(a) + f(b) is a good, if manual, way to convince oneself that such functions do exist and that this is the defining characteristic of the logarithm (and not just a convenient property). For example, log probability is central to information theory and thus many ML topics, and the fundamental reason is because Claude Shannon wanted a transformation on top of probability (self-information) that would turn the probability of multiple events into an addition — the aforementioned "f" is the transformation that fits this additive property (and a few others), hence log() everywhere. Interestingly, the logarithm “algorithm” was considered quite groundbreaking at the time; Johannes Kepler, a primary beneficiary of the breakthrough, dedicated one of his books to Napier. R. C. Pierce wrote: > Indeed, it has been postulated that logarithms literally lengthened the life spans of astronomers, who had formerly been sorely bent and often broken early by the masses of calculations their art required.

JackFr 2y ago

This is how I learned them in middle school — just common logs, as an aid to doing roots, powers and multiplications of big numbers. We were told in an off-hand way that logs could be to any base, even ‘e’, but not to worry about that for a few years.

meindnoch 2y ago

By the way, there's another function that can be used to turn multiplication into addition: f(x) = x^2 / 2 a * b = f(a + b) - (f(a) + f(b))

secondcoming 2y ago
Isn’t x^2 a multiplication?
- meindnoch 2y ago
  No, you misunderstood what I meant. Normally, a sliderule at distance x has the value of log(x) written on it, which allows doing multiplications by moving along the sliderule, since log(ab) = log(a) + log(b). Now imagine a sliderule onto which values of x^2/2 are written. This also allows you to multiply two numbers, because ab = (a+b)^2/2 - (a^2/2 + b^2/2).

pipes 2y ago

Coincidentally I watched this last night https://m.youtube.com/watch?v=7TWKSMtKCmU https://m.youtube.com/watch?v=7TWKSMtKCmU It gives the history / motivation behind logarithms and suddenly it became so much clearer to me. Pretty much multipling huge numbers by adding exponents , well I think I've understood that correctly? I think why I'm so interested in programming and computing is because I fascinated by the history of it all. It somehow acts as a motivation to understand it.

tim333 2y ago

I rather like Feynman's approach in the lecture Algebra from the Feynman Lectures https://www.feynmanlectures.caltech.edu/I_22.html https://www.feynmanlectures.caltech.edu/I_22.html He covers the inverse of the exponential, Henry Briggs' log tables and goes on to e^ix = cos x + i sin x The audio is also available https://www.feynmanlectures.caltech.edu/flptapes.html https://www.feynmanlectures.caltech.edu/flptapes.html

xiande04 2y ago

If you like this approach, I highly recommend Mathematics: It's Content, Methods, and Meaning by Kolmogorov. He uses this same approach, but applies it to many more concepts in math (about 1,000 pages!). In fact, I think I actually heard about that book on this site, so I guess I'm paying it forward. This approach was to align with the Soviet philosophy of dialectical materialism, which claims that all things arise from a material need. Not sure I'm fully onboard with the philosophy as a whole, but Kolmogorov's book was really eye opening.