The Lost Art of Logarithms

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?

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!<p>Lately, I&#x27;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 editor for initial drafts of projects I&#x27;m coding.<p>Does HN have love for any analog tools in particular?<p>[0]:<a href="https:&#x2F;&#x2F;sliderulemuseum.com&#x2F;" rel="nofollow">https:&#x2F;&#x2F;sliderulemuseum.com&#x2F;</a>

Interesting insight why applying a log transform often makes data normally distributed: Pretty much all laws of nature are multiplications (F=m<i>a, P</i>V=n<i>R</i>T, etc). If you start with i.i.d random variables and multiply them, you get log-normal data by virtue of the central limit theorem (because multiplications are additions on a log scale; and the CLT is also somewhat robust to non iid-ness). Thinking of data as the result of a lot of multiplications of influential factors, we thus get a log-normal distribution.

Here&#x27;s an logarithmic fact that I&#x27;ve made use of frequently:<p>If <i>X</i> is a random variable having a uniform distribution between zero and one, then –ln(<i>X</i>)&#x2F;<i>λ</i> has an exponential distribution with rate <i>λ</i>.<p>This relationship comes in handy when, for example, you want to draw weighted random samples. Or generating event times for simulations.

I started using LMAX Disruptor for some projects. One quirk with Disruptor is that the queue size always has to be an exponent of two.<p>I wanted to make sure that I always have at least enough room for any size and I didn&#x27;t want to manually compute, so I wrote this:<p><pre><code> var actualSize = Double.valueOf(Math.pow(2, Math.ceil(Math.log(approxSize) &#x2F; Math.log(2)))).intValue(); </code></pre> A bit much for a single line, but just using some basic log rules in order to the correct exponent. I learned all this in high school, but some of my coworkers thought I was using this amazing, arcane bit of math that had never been seen before. I guess they never use log outside of Big-O notation.

I can strongly recommend memorising some logarithms for use in mental maths. It&#x27;s given me powers I did not expect to have! Here&#x27;s what I wrote about it when I started: <a href="https:&#x2F;&#x2F;entropicthoughts.com&#x2F;learning-some-logarithms" rel="nofollow">https:&#x2F;&#x2F;entropicthoughts.com&#x2F;learning-some-logarithms</a>

The Logarithmic derivative is also something that is surprisingly fundamental.<p>(ln(f))&#x27; = f&#x27;&#x2F;f<p>In function theory you use it all the time. But people rarely notice, that it related to a logarithm.<p>Also the functions that have nice logarithmic derivative are a lot more interesting than expected. Nature is full of Gompertz functions. Once you&#x27;re familiar with it, you see it everywhere.

There used to be practical value to be able to do some basic back of the envelope log calculations in your head (no calculators, this was how you did fast multiplications&#x2F;divisions or exponents). There&#x27;s a story in Feyman&#x27;s Surely you&#x27;re joking book about Los Alamos scientists doing speed competitions for mental log calculations

(I&#x27;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

I learned the multiplication using addition and a lookup table in a class taught by Huffman (of Huffman compression fame). You weren&#x27;t allowed to use a calculator on the test.<p>But my absolute favorite trick is base conversions, <a href="https:&#x2F;&#x2F;www.khanacademy.org&#x2F;math&#x2F;algebra2&#x2F;x2ec2f6f830c9fb89:logs&#x2F;x2ec2f6f830c9fb89:change-of-base&#x2F;a&#x2F;logarithm-change-of-base-rule-intro" rel="nofollow">https:&#x2F;&#x2F;www.khanacademy.org&#x2F;math&#x2F;algebra2&#x2F;x2ec2f6f830c9fb89:...</a> with some practice you can do approximate base conversions (power to 2 to power of 10 or e) in your head

A 300 year old log table! What an opportunity to confirm Benford’s Law!<p><a href="https:&#x2F;&#x2F;en.m.wikipedia.org&#x2F;wiki&#x2F;Benford&#x27;s_law" rel="nofollow">https:&#x2F;&#x2F;en.m.wikipedia.org&#x2F;wiki&#x2F;Benford&#x27;s_law</a><p>“The discovery of Benford&#x27;s law goes back to 1881, when the Canadian-American astronomer Simon Newcomb noticed that in logarithm tables the earlier pages (that started with 1) were much more worn than the other pages.”

One of my favorite tricks in elementary school was to convince people I can calculate any logarithm for any number of their choosing.<p>&gt; Me: Pick any number.<p>&gt; Friend: Ok, 149,135,151<p>&gt; Me: The log is 8.2<p>Of course I&#x27;m simply counting the number of digits, using 10 as the base, and guessing the last decimal point, but it certainly impressed everyone.

Related: in a reaction to a comment I wrote about logarithms about a month ago[0], saulpw recently linked his own idea of making logarithms more &quot;accessible&quot; to the masses by introducing magnitude-based notation:<p><a href="https:&#x2F;&#x2F;saul.pw&#x2F;mag&#x2F;" rel="nofollow">https:&#x2F;&#x2F;saul.pw&#x2F;mag&#x2F;</a><p>I think it is a really nice idea that should be spread more widely. It might be Pi day, and while I traditionally complain that Tau is better for contrarian reasons (hey at least I&#x27;m honest), we might as well co-opt the extra attention maths gets for other mathematical causes.<p>[0] <a href="https:&#x2F;&#x2F;news.ycombinator.com&#x2F;item?id=43036094">https:&#x2F;&#x2F;news.ycombinator.com&#x2F;item?id=43036094</a>

Yeah love this angle - I made a video in a similar vein: <a href="https:&#x2F;&#x2F;www.youtube.com&#x2F;watch?v=OjIwCOevUew" rel="nofollow">https:&#x2F;&#x2F;www.youtube.com&#x2F;watch?v=OjIwCOevUew</a>

&gt; Charles Petzold<p>Haven&#x27;t heard that name in a while. For me he&#x27;s the WinApi guy - learned a lot from him when I first started programming.

Notes from &quot;How should logarithms be taught?&quot; (2021) <a href="https:&#x2F;&#x2F;news.ycombinator.com&#x2F;item?id=28519356">https:&#x2F;&#x2F;news.ycombinator.com&#x2F;item?id=28519356</a> re: logarithms in the Python standard library, NumPy, SymPy, TensorFlow, PyTorch, Wikipedia

Charles Petzold was one of my favorite tech authors from the way begone era. Written in a style very similar in vein to the Lost of Art of Logarithms he made me fall in love with the various mundane tech concepts that would never jump out as a anything of interest, otherwise. What a treat!

The traditional explanations of logarithms I&#x27;ve encountered are far too math and terminology heavy for most people to grasp.<p>Think of a number line.... show example..... 1..2..3..4..5.. etc<p>Any given move to the right, makes the value go up by 1.<p>But... What if we did a <i>special number line</i> where each time it doubled instead of adding one? 1..2..4..8..16, etc...<p>That line would go up way to fast to see numbers like 10, so we can expand it out a bit...show that... and start to fill in the numbers... 2^10 (1024) is almost 1000... so you can divide that distance by 3 to get 10 on the line, then move one unit left for 5... and build out a slide ruler.<p>Computing logarithms with a 4 function calculator isn&#x27;t hard by the way, I used to do it for fun on my lunch breaks.

I recommend the classic <i>Introduction to Logarithms</i>, by Cormen, Rivest, Leiserson et al.

Notation for writing log has always bugged me. Like I feel like it should be more like &lt;10&gt;^&lt;527&gt; which would be the log base 10 of 527. That&#x27;s not it, but something. The current notation just doesn&#x27;t feel quite right.

If the author is in here, thank you. Been looking for a text for my daughter on the subject. This might just fit the bill. If you&#x27;re just the linker, then thank you Ozanonay.

Off-topic, but anyone knows where to buy such [1] old but not rare (so they are cheap enough) math books? In UK and globally? Is e-bay and perhaps amazon the best place? How to avoid fakes?<p>[1] <a href="https:&#x2F;&#x2F;www.lostartoflogarithms.com&#x2F;chapter01&#x2F;" rel="nofollow">https:&#x2F;&#x2F;www.lostartoflogarithms.com&#x2F;chapter01&#x2F;</a>

Is this the same author who wrote Win32 API books?

I have a few old math manuals at home, from late 19th &#x2F; early 20th century. Many of them have a logarithm table as an appendix. It looked like the type of things that if you had a few extra sheets to print to make a booklet, you&#x27;d just add because it was bound to be very useful to someone.

I wish there was a mailing list I could subscribe to so I could know when the book os complete.<p>Or a pre-order on Amazon?

One of the best uses of logarithms is how they can be used to quickly calculate db (as in decibel) gains and losses mentally. See this older comment for more details<p><a href="https:&#x2F;&#x2F;news.ycombinator.com&#x2F;item?id=32550539">https:&#x2F;&#x2F;news.ycombinator.com&#x2F;item?id=32550539</a>

Wow, I thought it was just some random guy, but was then quite surprised to see this was being authored by none other than the legendary Charles Petzold. I&#x27;d buy this book - just to put it next to my copy of &quot;Programming Windows 95&quot; (who remembers?) :)

There was an interesting text, by Isaac Asimov, where he explained in a very clear way the historical importance of logarithms -- they allowed Kepler to finalize his work by replacing tables of multiplications (which were difficult and error-prone) with sums.

I feel frustrated that we cannot conceive of numbers like 10^80 (atoms in the universe) or 10^4000 (number configurations for a system with 4000 variables having 10 states each). Maybe there are superbrains out there in the universe that can do so.

Funny how logarithms shaped navigation, astronomy, and music—yet now they&#x27;re mostly a forgotten button gathering dust on our calculators. Hidden tech history right there.

Logging the World - Oliver Johnson (Oxford Mathematics): <a href="https:&#x2F;&#x2F;youtu.be&#x2F;UsK52iZMsxo" rel="nofollow">https:&#x2F;&#x2F;youtu.be&#x2F;UsK52iZMsxo</a>

So interesting! The author doesn’t, I believe, yet cover how the first log tables were computed (by hand), so I asked ChatGPT. This may be of interest: <a href="https:&#x2F;&#x2F;chatgpt.com&#x2F;share&#x2F;67d3a64d-f8a8-8012-bde3-e80813b2b402" rel="nofollow">https:&#x2F;&#x2F;chatgpt.com&#x2F;share&#x2F;67d3a64d-f8a8-8012-bde3-e80813b2b4...</a>

Well written and fun to read! (For nerds)

Charles Petzold wrote one of my favorite books - &quot;Code: The Hidden Language of Computer Hardware and Software&quot;. Very excited to see how this turns out and thanks for giving some of this knowledge away for free!