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The Lost Art of Logarithms

lostartoflogarithms.com

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Re: The Lost Art of Logarithms

#111

Post by ted_dunning »

BinRoo wrote: post #69

One of my favorite tricks in elementary school was to convince people I can calculate any logarithm for any number of their choosing. > Me: Pick any number. > Friend: Ok, 149,135,151 > Me: The log is 8.2 Of course I'm simply counting the number of digits, using 10 as the base, and guessing the last decimal point, but it certainly impressed everyone.

You can do even better if you memorize three numbers: 301, 477, 845. These are the values of 1000log10(n) for n = 2, 3, 7. From these you can quickly get the values for 4 (= 22), 5 (=10/2), 6 (=23), 8 (=222) and 9 (=33).

For your example 1.49 is close to 3 / 2 so the log will be very close 0.477 - 0.301 = 0.176.

This means that your answer is near 8.176 (actual value is 8.173).

This tiny table of logs can also let you answer parlor trick questions like what is the first digit of 2^1000 (the result is very nearly 10^301 but a bit above, so 1 is the leading digit).

Re: The Lost Art of Logarithms

#112

Post by BobaFloutist »

II2II wrote: post #107
agumonkey wrote: post #61

Earlier quoted context omitted.

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.

I think it is a combination of factors. Mathematical pedagogy is legitimate if the end goal is to train mathematicians, so yes it is geared towards those who think in the abstract. (I'm going to ignore the comment about very smart, since I don't think mathematical ability should be used as a proxy for intelligence.) On the other side, I don't think those who are involved in curriculum development are very skilled in…

Frankly I wish I had known integral calculus going into geometry, I could tell there was a pattern behind formulas for areas and volumes but I couldn't for the life of me figure it out. There are worse ways to remember the formula for the volume of a sphere than banging out a quick integral!

Re: The Lost Art of Logarithms

#113

Post by BobaFloutist »

kccqzy wrote: post #89
dkislyuk wrote: post #60

Earlier quoted context omitted.

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.

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.

You see how one of those isn't like the others?

Re: The Lost Art of Logarithms

#114

Post by dkislyuk »

coffeemug wrote: post #102
dkislyuk wrote: post #45

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 lo…

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

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.

Re: The Lost Art of Logarithms

#115

Post by analog31 »

zwnow wrote: post #24
tmoertel wrote: post #20

Here's an logarithmic fact that I've made use of frequently: If X is a random variable having a uniform distribution between zero and one, then –ln( X )/ λ has an exponential distribution with rate λ . This relationship comes in handy when, for example, you want to draw weighted random samples. Or generating event times for simulations.

How long do I have to study math to understand this?

It seems like there's always more than one way to skin a cat, but I'd have turned to calculus, had I needed to derive something like this and didn't think to look it up (e.g., before the Internet).

Re: The Lost Art of Logarithms

#117

Post by xelxebar »

Rendello wrote: post #105
xelxebar wrote: post #94

How timely! I just learned how to use a slide rule yesterday. Looking to pick one up, and a bit overwhelmed by the plethora of choices, I went down a small rabbit hole[0]. Some slide rules produced are pure works of art! Lately, I've been rediscovering the surprising niceties that analog tools can provide over our everything-is-a-panel-of-glass interfaces these days. Recently, I have been enjoying pen and paper as my…

I've been doing a math course and occasionally think of picking up these analogue tools. Someone on Hacker News had me interested in the Soroban, the Japanese abacus [1], which is still used to train insane mental math speeds to this day [2]. 1. https://en.wikipedia.org/wiki/Soroban 2. https://www.youtube.com/watch?v=s6OmqXCsYt8

We're definitely on a similar wavelength. I actually own a couple Japanese abaci and know the basics. Top performers feel near magical: double-fisters[0] and blazingly fast mental arithmetic [1].

[0]:https://www.youtube.com/watch?v=EK6uIjjkrGE

[1]:https://youtu.be/-kjUCtqSWlw?feature=shared&t=451

Re: The Lost Art of Logarithms

#118

Post by mixmastamyk »

mixmastamyk wrote: post #88
sva_ wrote: post #50

> Charles Petzold Haven't heard that name in a while. For me he's the WinApi guy - learned a lot from him when I first started programming.

Still can! His classic book Code is fantastic and has a recent second addition.

^edition.

Re: The Lost Art of Logarithms

#119

Post by floydnoel »

dekhn wrote: post #38
inasio wrote: post #34

(I'm sure this is in the book) John Napier, the father of logarithms (the N in ln), basically had a sweatshop of human calculators making log tables over something like 20 years - critical for celestial navigation. There was a huge price attached to the person that developed a method to safely navigate across the oceans, also lead to the invention of the pocket watch

isn't the n in ln "natural" ("logarithm natural")?

yes i was taught that "ln" stood for "natural log"

would be interested to hear other definitions!

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