[rramadass]

A better way to understand logarithms is to start with the original motivation from Napier himself (https://sites.pitt.edu/~super1/lecture/lec44911/005.htm);

Seeing there is nothing (right well-beloved Students of the Mathematics) that is so troublesome to mathematical practice, nor that doth more molest and hinder calculators, than the multiplications, divisions, square and cubical extractions of great numbers, which besides the tedious expense of time are for the most part subject to many slippery errors, I began therefore to consider in my mind by what certain and ready art I might remove those hindrances. And having thought upon many things to this purpose, I found at length some excellent brief rules to be treated of (perhaps) hereafter. But amongst all, none more profitable than this which together with the hard and tedious multiplications, divisions, and extractions of roots, doth also cast away from the work itself even the very numbers themselves that are to be multiplied, divided and resolved into roots, and putteth other numbers in their place which perform as much as they can do, only by addition and subtraction, division by two or division by three.

This is what provides the intuition viz; convert multiplication/division/etc. of large numbers into addition/subtraction of two other smaller numbers. Logarithms as inverse of Exponentiation came much later. Starting with this generally confuses the student since they do not understand the point of it all.

From https://en.wikipedia.org/wiki/History_of_logarithms;

Napier conceived the logarithm as the relationship between two particles moving along a line, one at constant speed and the other at a speed proportional to its distance from a fixed endpoint.

Since the speed is directly proportional to its remaining distance from the fixed endpoint, it therefore is a deceleration, which results in the characteristic "flattening" of the curve.

Further details for understanding the above can be found at Priority, Parallel Discovery, and Pre-eminence: Napier, Burgi and the Early History of the Logarithm Relation (pdf) - http://www.numdam.org/item/RHM_2012__18_2_223_0.pdf

[Mr_Minderbinder]

I find it surprising that logarithms and e (a.k.a. Napier’s constant), were developed and discovered only relatively recently in the history of mathematics despite how natural and fundamental they are.

The idea of exponential growth and the practice of charging interest in finance are both ancient. Surely an ancient mathematician would have investigated these in depth and discovered what Napier, Bernoulli and others found?

A while ago I was solving several infinite series of exponentials in the context of a problem concerning the half-life of medicines and I made frequent use of logarithms. That is when I started to wonder about their history.

[rramadass]

Arithmetic series/Geometric series/Mathematical Tables (construction and lookup) were all well known in the ancient world. Wikipedia mentions Babylonians and Indians for logarithms (https://en.wikipedia.org/wiki/History_of_logarithms#Predeces...) but doubtless others too had variations of the same. Travel/Trade/Finance/Astronomy/Astrology would have been the main drivers before the modern scientific era.

Here is an interesting book; The History of Mathematical Tables: from Sumer to Spreadsheets - https://en.wikipedia.org/wiki/The_History_of_Mathematical_Ta...

We have lost a lot of knowledge accumulated and written down on biodegradable material (eg. papyri/palm leaves etc.) before the advent of the printing press made knowledge dissemination cheaper and easier. We then compounded the problem by dismissing everything before the beginning of our "scientific era" as being primitive/superstitious/non-methodical/etc. Only later on did we realize that many ancient civilizations were quite advanced in many aspects of mathematics and science though their way of approaching/inventing/recording was quite different from our "modern scientific method" and therefore we need to research them from a different pov and without condescension.

PS: History of Mathematics - https://en.wikipedia.org/wiki/History_of_mathematics

[jongjong]

I find my explanation simpler.

// The power to which I must raise 10 to get 100 is 2.

log10(100) = 2

// The power to which I must raise 10 to get 1000 is 3.

log10(1000) = 3

// The power to which I must raise 3 to get 27 is 3.

log3(27) = 3

Also it makes solving equations much more intuitive:

log3(x) = 4

^ This means; the power to which I must raise 3 to get x is 4. So it follows logically that if I raise 3 to the power of 4, I will get x. This makes it intuitive that this equation can be rewritten as:

x = 3 ^ 4

You don't even need to know the algebraic rule. I felt retarded when I figured this out. This was a rule I had memorized before. It's even dumber and easier to infer than the rule to compute derivatives. I wonder why teachers even bother to teach you all these rules when they could just explain the fundamentals to you.

[rramadass]

That is just the definition of Logarithm which is what is taught to all students today i.e.

Given a^x = b we define log_a(b) = x where 'a' is a +ve real number - https://en.wikipedia.org/wiki/Logarithm#Definition

The above wikipedia page also details the properties, applications and generalization of the logarithm concept which are non-trivial.

As i pointed out above, that does not help in intuiting why it is helpful and needed. That is why you need to read the history of logarithms and see how we arrived at the above standard.

Napier actually calculated logarithms of sines for every minute from 0-90degrees to simplify astronomical calculations. The complexity/sizes involved, precision needed etc. can all be seen in this detailed paper walking you through the entire process of table construction; Napier’s ideal construction of the logarithms (pdf) - https://locomat.loria.fr/napier/napier1619construction.pdf

[jongjong]

I think part of the confusion is that the base isn't always clear. A lot of times, when first exposed to logs, we are shown only base 10 or base e so it's just written as the word log or ln... This omission obscures the meaning and versatility of the function. Log and ln are essentially presented as separate constructs. Which is retarded.

Personally I only remember being briefly exposed to log and ln in school. I don't recall doing any complex algebra with them. So when I started doing algebra with them at university, I had to kind of re-learn from scratch.

I asked Claude about the weird way I was taught logs and it said this, which I found interesting:

There's a broader pattern here that shows up all over maths: notation and procedures get taught as the primary object, with the meaning treated as optional enrichment that "advanced" students might appreciate later. For a lot of people it's backwards — the meaning is the cheap, load-bearing thing, and the procedures are what should be derived from it. Logs are maybe the most egregant case because the gap between "incomprehensible button" and "obvious once stated" is a single plain-English sentence.

[rramadass]

Quotes from Richard Hamming;

Mathematics is an interesting intellectual sport but it should not be allowed to stand in the way of obtaining sensible information about physical processes.

The purpose of computing [i.e. using mathematics] is insight, not numbers.

Does anyone believe that the difference between the Lebesgue and Riemann integrals can have physical significance, and that whether say, an airplane would or would not fly could depend on this difference? If such were claimed, I should not care to fly in that plane.

What this implies is that Mathematics is merely a formalism i.e. set of symbols/operations/rules manipulated using logic. Think of it as just a language which is succinct, precise and standardized. Only when it explains something in the physical world does it have meaning else it is merely a game.

Quotes from Albert Einstein;

Imagination is more important than knowledge. Knowledge is limited. Imagination encircles the world.

Logic will get you from A to Z; imagination will get you everywhere.

Any fool can know. The point is to understand.

The mere formulation of a problem is far more essential than its solution, which may be merely a matter of mathematical or experimental skills. To raise new questions, new possibilities, to regard old problems from a new angle requires creative imagination and marks real advances in science.

If I had an hour to solve a problem, I'd spend 55 minutes thinking about the problem and five minutes thinking about solutions.

What this implies is that it is important to exercise imagination to understand different perspectives of a problem eg. historical/existing/new ones, and reformulate it properly before focusing on mere formalisms.

In the 1960s the "New Math" movement was instituted to teach "Mathematics the Modern Way" which though criticized, took root and has permeated mathematics education to this day to the detriment of the students and the field - https://en.wikipedia.org/wiki/New_Math Read the "Reception" and "Legacy" sections in particular.

So to understand mathematics, look to history when a particular concept first came into existence to explain/solve a particular problem and then look at the formalism standardized/employed today to generalize it. Now we can understand the motivation behind the current formalism better and its applications to the real world. This is called the "Genetic Method" of teaching mathematics - https://en.wikipedia.org/wiki/Genetic_method

For example; logarithms were invented by Napier long before exponentiation was formally defined i.e. he had no concept of a "base"! The resources i had linked to above detail his construction. So what does it mean when today we define logarithm as inverse of exponentiation?

It means logarithm is merely a table lookup from one set of numbers to another where one is a geometric series and the other a arithmetic series and its sole motivation was to map multiplication/division/etc. of large numbers onto addition/subtraction of smaller numbers. In our current mathematics we have just formalized it using the language of functions i.e. exponentiation/logarithms.