Glossary / Logarithm
An ordinary sheet of paper is 0.1 mm thick. Fold it in half, then again, and again. The Moon is 384,400 km away, so it seems you would need thousands or even millions of folds. Intuition is badly wrong here. A logarithm is exactly the answer to this kind of question: how many times do you have to double to get from one number to another.
“In plain words” is hidden on all puzzles.
Move the slider. Each fold lays the sheet on top of itself, so the thickness doubles: the lower part of the column is what you had before the fold, the upper part is an identical copy on top. Next to it, in the same scale, are familiar things. How many folds will you need to reach the Moon?
How many folds it takes to outgrow each thing:
In reality a sheet can only be folded about twelve times: after that it is too thick and stiff. But the arithmetic of doubling stays the same.
Look at the steps under the chart. The things differ by factors of thousands and millions, yet the fold numbers go up calmly: 9, 15, 27, 32, 42. That is exactly what a logarithm does: it turns "how many times bigger" into "how many doubling steps". From a book to Everest the ratio is almost three hundred thousand, yet it takes only 18 folds.
“Rigorous” is hidden on all puzzles.
The logarithm of x to base b is how many times you have to multiply b by itself to get x. It is written logb x. The most common bases are 2 (doubling) and 10 (counting zeros).
2 · 2 · 2 = 8 → log2 8 = 3
210 = 1024 → log2 1024 = 10
For numbers between powers the logarithm is a fraction: log2 10 ≈ 3.32, because 10 lies between 8 and 16.
10 · 10 · 10 = 1000 → log10 1000 = 3
The base-10 logarithm is almost the number of digits: for a number with d digits it lies between d − 1 and d. A million has 7 digits, and log10 1,000,000 = 6.
How many sheet thicknesses fit between us and the Moon? 384,400 km = 384,400,000,000 mm = 3,844,000,000,000 times 0.1 mm. Now ask: how many times do you double 1 to get this number?
log2 3,844,000,000,000 ≈ 41.8 → it takes 42 folds
After 41 folds the sheet still falls short; after 42 it is already thicker than the distance to the Moon: 439,805 km.
The most useful fact about logarithms for mental arithmetic: 210 = 1024, almost 1000. So every ten doublings add roughly three zeros to a number. This is also why a kilobyte is 1024 bytes.
| Folds | Factor | Thickness |
|---|---|---|
| 10 | ≈ a thousand | 10.2 cm |
| 20 | ≈ a million | 105 m |
| 30 | ≈ a billion | 107 km |
| 40 | ≈ a trillion | 109,951 km |
The factor is a little bigger than the round number: 1024 instead of 1000, so the thickness always gets a bit ahead of the round figure.
Here is the thickness of the folded sheet after each fold, and next to it an ordinary stack where one more sheet is added each time. On an ordinary scale almost all the folds stick to zero, then the curve shoots up. Switch to the logarithmic one: each mark there is ten times the previous one, and doubling becomes a straight line.
Many familiar units are really logarithms: one step on the scale means not "plus this much" but "this many times".
“Deeper” is hidden on all puzzles.
bm · bn = bm + n ⇒ log (a · c) = log a + log c
Multiplying powers means adding exponents. So the logarithm of a product is the sum of the logarithms. John Napier published tables of logarithms in 1614, and astronomers began adding numbers from tables instead of doing long multiplication. Soon the slide rule appeared: if numbers are marked at distance log x from the start, laying one segment after another multiplies the numbers. Engineers used it for centuries, right up to pocket calculators.
–
The natural logarithm ln is used here, but the base doesn't matter: any base stretches all lengths by the same factor, and the two segments still add up to the product.
logb x = loga x / loga b
Logarithms to different bases differ only by a constant factor. For example, log2 x ≈ 3.32 · log10 x. A million has 6 zeros, so it takes 6 · 3.32 ≈ 19.9, that is 20 doublings to reach it. That is why a chart on a log scale looks the same for any base: only the labels on the marks change.
Put one dollar in at 100% a year. Compounded once a year, you get 2 dollars. Twice at 50%, 2.25. Monthly, 2.61; daily, 2.71. The more often, the closer to a limit that stops growing: e ≈ 2.71828. It is the natural base of continuous growth, and the logarithm to it, ln, is called natural.
(1 + 1/n)n → e as n → ∞ · ln 2 ≈ 0,693
From ln 2 ≈ 0.693 comes the "rule of 70": at r% growth a year, an amount doubles in about 70 / r years. Check how accurate it is.
A logarithm also works the other way round. Think of a number from 1 to N. Each answer to "is it at most ...?", asked about the middle of the range, throws away half the options. After k questions N / 2k remain, so it takes about log2 N questions. This is how you find a word in a dictionary or a row in a sorted table.
Each row is one question. The bar length is the logarithm of how many numbers are still left. That is why the bars shrink in equal steps: halving on a log scale is a straight line again.
The middle value: line everyone up in order, and half are below it, half above.
A millionaire walks into a bar: the average income shoots up, the median barely moves.
The sum of all values divided by how many there are: what everyone would get if it were shared equally.
Almost everyone has more legs than average: the average is a little under two.
The average result you can expect if you repeat the same thing very many times.
A die's expected value is 3.5, yet no face shows 3.5.
A number from −1 to +1 showing how closely two quantities move together.
In summer both ice cream sales and drownings go up. Ice cream isn't to blame: the heat drives both.
A measure of spread: how far values typically stray from the mean. The standard deviation is its square root, in the same units.
Two towns with the same average temperature: one stays near +15 all year, the other swings from −20 to +50.
A bell-shaped distribution: most values sit near the mean, and the further away, the rarer.
Every ball on a Galton board falls at random, yet thousands of balls always build the same bell.
The more repetitions, the closer the share of an outcome gets to its true probability.
The share of heads creeps towards 50%, while the gap between heads and tails in coins usually grows.
The chance of an event given that something else is already known.
Almost all NBA players are tall, but almost all tall people are not NBA players.
Events are independent if knowing about one doesn't change the chances of the other.
After five heads in a row tails is no more likely: the coin remembers nothing.
A sample some people get into more easily than others, so it describes everyone badly.
In 1936 a magazine polled millions of Americans and still picked the wrong election winner.
How often chance alone would produce a gap at least as big as yours if there were really no difference.
When there is no effect at all, p below 0.05 turns up as often as p between 0.50 and 0.55.
A range around a measured number, built by a method that covers the true value in a chosen share of samples, such as 95%.
Of twenty honest 95% intervals, on average one misses the truth, and none of them knows whether it is the one.
The probability that a test notices a real effect of a given size. A power of 80% means the test misses one working effect in five.
A weak A/B test that does find an effect almost always exaggerates it: a real +10% shows up on the dashboard as +24%.
How far an average or a rate computed from a sample usually misses the truth: σ divided by the square root of the sample size.
Halving the error takes four times as much data.