What Is a Logarithm

Most people were taught logarithms as a button on a calculator and a rule to memorise. That is why they do not stick. A logarithm is not a rule. It is an answer to a question, and the question is very old and very practical.

Start with the question

Multiplication is hard. Addition is easy.

That sounds trivial until you have to do it by hand. Multiply 4,096 by 32,768 with a pencil and you will be at it for a minute and you will probably get it wrong. Add two numbers of the same size and you are done in seconds.

So here is the question somebody asked four hundred years ago: is there a way to turn every multiplication into an addition?

There is. That is what a logarithm is for. Everything else about them is detail.

What it is

A logarithm asks: what power do I raise this base to, in order to get that number?

So log₂(8) = 3, because 2³ = 8. And log₁₀(1000) = 3, because 10³ = 1000. If you can read an exponent backwards, you can read a logarithm.

A logarithm is just an exponent that has been pulled out and looked at on its own. The moment you hold that, the rest follows without memorisation.

The one property that matters

Exponents add when you multiply the things they sit on. 2² × 2³ = 2⁵, because you are stacking two twos next to three twos and counting five. That is not a rule to learn — it is what an exponent means.

Pull the exponents out and look at them alone, and you get:

That single line is the reason logarithms exist. John Napier published the first tables in 1614 for exactly this: astronomers were spending their lives multiplying enormous numbers, and he handed them a way to look up two values, add them, and look the answer back up. Laplace later said logarithms had, by shortening the labours, doubled the life of the astronomer.

Now the part almost nobody is taught

Here is a question worth sitting with. Is the logarithm the only function that does this?

Suppose you want some function f that converts products into sums — so that f(ab) = f(a) + f(b) for every positive a and b. Suppose you also want it to be continuous, meaning no sudden jumps.

It turns out those two requirements pin it down completely. The only continuous functions satisfying that equation are the logarithms — f(x) = c · ln(x), for some constant c. This is a standard result, a version of Cauchy’s functional equation, and it is the most important thing on this page.

The base almost never matters

You will see three bases in the wild. Base 10 (written log), base 2 (used in computing and information), and base e ≈ 2.71828, called the natural logarithm and written ln.

Changing between them is a single multiplication:

So the choice of base changes your units and nothing else — the way choosing feet instead of metres changes the number on the tape measure and not the length of the room. If an argument depends on which base you picked, the argument is broken.

Compression: the property you have already used

Because equal ratios become equal steps, a logarithm takes a range that spans orders of magnitude and folds it into something a human can hold.

Figure 01

Ten thousand becomes four

10

101

1002

1,0003

10,0004

bar length = the raw numberright column = its base-10 logarithm

This is why so many everyday scales are logarithmic, and you have been using them without being told:

• Decibels — sound intensity
• Richter — earthquake amplitude
• pH — acidity, as −log of concentration
• Stellar magnitude — brightness
• Octaves — each one doubles the frequency

An earthquake of magnitude 7 is not slightly worse than a 6. It is ten times the amplitude. The scale adds because the world multiplies.

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Now put it to work

Everything above is standard. Here is where it earns its place in my own work, and where I think the logarithm stops being a technique and becomes a small piece of philosophy.

I hold that what you experience is not what happened. It is a ratio:

Now watch what that forces.

1. Reality is a ratio. Not a difference, not a total — a quotient of what arrived over what was expected.
2. Ratios combine by multiplying. A day that went twice as well as expected, followed by one that went three times as well, is six times — not five.
3. But experience combines by adding. You do not feel Tuesday times Wednesday. Episodes accumulate. A week is a sum of its days, not a product of them.
4. So something must convert multiplication into addition — and by the result above, only one kind of function can.

That is the argument, and I want to be clear about what it does and does not establish. It does not prove that experience works this way. It proves that if Reality is a ratio and if felt episodes add, then the log is the only bridge available. The two ifs are doing real work, and the second one is testable — I will come to that.

What the numbers look like

Three things fall out of that table, and they are the reason I use a logarithm rather than a subtraction.

One — zero is where it should be

When the actual matches the expectation, R = 1, and ln(1) = 0. No surprise, no signal. The origin of the felt scale lands exactly on “as expected” without anyone having to place it there. A subtraction would have needed you to choose a reference point. The log chooses it for you.

Two — the sign is the direction

Below expectation gives a negative number. Above gives a positive one. Disappointment and delight are not two different quantities. They are one quantity with two signs.

Three — and this is the one worth arguing about

Look at the symmetry. Twice what you expected is +0.693. Half what you expected is −0.693. Exactly equal and opposite.

Now try it with subtraction, the obvious alternative. Getting 2 when you expected 1 is a difference of +1. Getting 0.5 when you expected 1 is a difference of −0.5. Twice as good and half as good come out lopsided — the good one counted double.

I think that matches how departure actually feels, and it is the strongest reason I know for preferring the ratio to the difference. But I think is not evidence, which brings us to the last section.

Figure 02

The shape of the felt scale

Am I right about this?

The claim that felt episodes add is not a definition. It is an empirical bet, and it has a history. In 1860 Gustav Fechner, building on Ernst Weber’s finding that the smallest noticeable change in a stimulus is proportional to the stimulus already present, proposed that sensation grows as the logarithm of intensity. That is the Weber–Fechner law, and it is the same shape as my equation arrived at from a different direction.

The experiment

Here is what would settle it, and it is an ordinary study that any psychology department could run.

1. Give people a sequence of episodes with a measurable actual and a pre-registered expectation — monetary outcomes against a stated forecast will do.
2. Collect a self-reported felt magnitude for each episode, and then for the sequence as a whole.
3. Compute two predictions of the whole from the parts: the sum of logs, and the sum of linear differences.
4. See which one tracks the reported total.

That is what a claim looks like when it is willing to be wrong. It is also the point of teaching you logarithms this way rather than as a calculator button: once you can read the function, you can see exactly where a theory built on it would break.

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Exercises

Do these on paper. The answers are hidden until you want them.

1. Read it backwards. What is log₂(32)? And log₁₀(0.01)?
   Answer

   5, because 2⁵ = 32. And −2, because 10⁻² = 0.01. Negative logarithms are how you say “smaller than one” — which is exactly the case R < 1 in the table above.

2. Use the property. Given ln(3) ≈ 1.099 and ln(4) ≈ 1.386, find ln(12) without a calculator.
   Answer

   2.485. Because 12 = 3 × 4, and the logarithm turns that product into the sum 1.099 + 1.386. This is precisely the trick Napier sold to astronomers.

3. Change the base. You have a calculator with only ln. Compute log₂(8).
   Answer

   ln(8) / ln(2) = 2.079 / 0.693 = 3. The base you divide by is the base you end up in.

4. Apply it. You expected 40 sales and got 10. Then you expected 10 and got 40. What is S in each case, and what does the pair tell you?
   Answer

   R = 0.25 gives S = −1.386; R = 4 gives S = +1.386. Equal and opposite. On a subtraction scale the two would be −30 and +30 — which happens to look symmetric here, but try 1 expected against 2 actual versus 2 expected against 1 actual and the subtraction gives +1 and −1 while the ratios give ±0.693. The log is symmetric for every pair; subtraction only sometimes.

5. Break it. Find a case where treating experience as additive is obviously wrong.
   Answer

   There are several, and I am not going to pretend otherwise. Trauma does not add — a single severe episode can dominate a whole year in a way no sum reproduces. Adaptation means a repeated departure stops registering. If you found one of these, you have found a real limit of the model, which is a better outcome for this lesson than getting the arithmetic right.

What to carry away

1. A logarithm is an exponent, pulled out and looked at on its own.
2. It turns multiplying into adding — and it is the only continuous function that does.
3. The base is a choice of units, never a choice of substance.
4. Equal ratios become equal steps, which is why it compresses and why zero lands on “as expected.”
5. If a quantity is a ratio and its effects accumulate, the logarithm is not optional. That is the whole reason my felt scale has one in it.