A Primer on
Log Space

The ruler

A logarithm is not a formula to memorise. It is a change of ruler. On an ordinary ruler, equal distances mean equal differences: the gap from 3 to 4 is the same as the gap from 103 to 104. On a log ruler, equal distances mean equal ratios: the gap from 3 to 6 is the same as the gap from 100 to 200, because both are a doubling.

That single swap is the whole subject. Everything below is a consequence of it.

The middle of two numbers moves

Ask someone for the number halfway between 2,000 and 10,000 and they will say 6,000. On a ratio ruler the answer is about 4,472, because that is the number sitting the same multiple above 2,000 as it does below 10,000 — roughly 2.24× in each direction.

The first answer is the arithmetic mean. The second is the geometric mean, and you have now met the main character. Drag the ends below and watch the two midpoints separate: they agree only when the two numbers are close, and diverge without limit as they spread.

Multiplying becomes adding

If equal distances are equal ratios, then placing two lengths end to end multiplies the ratios they represent. Move three doublings along the ruler and you have multiplied by eight. This is exactly what a slide rule did, and why one was enough to design aircraft with.

Decibels, octaves, decades, decibans

A log needs a base, and the choice is only ever about what size of step you want to call one. Every unit below is the same idea wearing different clothes.

Two conversions are worth keeping in your head, because they turn most mental arithmetic in this area into addition: ×2 is +3 dB and ×10 is +10 dB. From those two you can read almost anything. A million is ×10 six times, so +60 dB. A thousand-fold is +30 dB. An eight-fold is three doublings, so +9 dB.

Counting doublings tames big numbers

How many times do you halve 1,048,576 to reach 1? Twenty. That is log₂, and it is why a million-point health bar is only twenty hits if each hit halves it. The number stops being frightening the moment you count doublings instead of units.

The same trick runs forwards. Anything growing at a steady percentage has a fixed doubling time, and you can get it in your head: divide 70 by the percentage. Five percent a second doubles every fourteen seconds. Seven percent a year doubles every decade.

What multiplying does to you

Now the consequences. Everything here follows from one fact: when a quantity is multiplied rather than added to, its behaviour over time is governed by the sum of the logs, and that is not the same thing as the average you were quoted.

Percentages do not reverse

Lose 50% and you need +100% to get back. Not +50%. The loss and the recovery are equal distances on the ratio ruler only when they are equal ratios, and −50% and +50% are not: one is ×0.5, the other is ×1.5.

Look at the third column. In log space the asymmetry vanishes: a −3 dB loss is repaired by a +3 dB gain, every time, with no special cases. The asymmetry was never in the world. It was in the units.

You cannot multiply your way to zero

Halve something forever and it never arrives at nothing. This sounds like a technicality and is not: it means ruin, for a multiplicative quantity, cannot be an endpoint. It has to be a threshold that somebody chooses.

It also means recovery from a deep hole is not merely hard, it is differently hard. Getting from ×0.01 back to ×1 is exactly as far as getting from ×1 to ×100.

The average you are quoted is not the rate you grow at

Here is the centre of everything. Consider a bet that pays +60% or −40%, each equally likely. The average outcome is (1.6 + 0.6) / 2 = 1.10, a tidy 10% a round, and that number is completely true.

Now take the bet twice, winning once and losing once. You have 1.6 × 0.6 = 0.96. You are down, on a bet that averages +10%. Take it two hundred times and the typical outcome is 0.98²⁰⁰, which is about ×0.02.

The gap has a name and an approximate size. It is called volatility drag, and to a good approximation it is half the variance.

Same average, tighter spread, more growth

Because drag depends on the spread and not the mean, you can improve an outcome without touching its average at all. Two bets that both average +10%:

Identical averages; one grows and one shrinks. This is why reducing volatility is a real improvement rather than an act of timidity, and why any summary that reports only a mean has thrown away the thing that decides the outcome.

One sequence, not many

The average across worlds is not the average across time

Expected value answers a specific question: if a great many people each took this bet once, what would the total come to? That is a real question with a real answer, and for a great many purposes it is the right one.

It is not the question you are asking when the thing compounds. You do not get a great many tries in parallel. You get one sequence, and the quantity that governs it is the growth rate along that sequence.

For the +60%/−40% bet, both answers are correct and they point in opposite directions. Across a thousand parallel players the total grows at 10% a round, driven almost entirely by a handful of extraordinary runs. Along any single path it shrinks at about 2% a round. Nobody is lying. They are answering different questions.

How much to stake

Once growth is the objective, "how much should I bet" has a real answer rather than a temperament. Stake a fraction f of what you have, on something with excess return μ and variance σ², and your growth rate is:

The return term is linear in the stake; the drag term is quadratic. So doubling your stake doubles the expected gain and quadruples the penalty. There is an optimum, and past it betting more makes you poorer while continuing to raise your expected value.

For a simple bet you win with probability p, the growth-maximising fraction is f* = 2p − 1. At 60/40 odds — a genuinely good edge — that is a fifth of your stack. Expected value, meanwhile, keeps climbing all the way to everything.

Diversification is a growth strategy, not a safety blanket

Spreading across things that do not move together lowers the spread of the whole without lowering its average — and by the previous section, that raises the rate you compound at. It is not a trade of return for comfort. It is a way to be paid more.

The sharpest version: an asset that loses money on its own can raise a portfolio's growth rate, if it is volatile and uncorrelated enough that it lifts the average more than it adds to the spread. This is not a trick. It follows from an asset having no growth rate that belongs to it — only an effect on the growth rate of everything held together.

The average is computed over what survived

Every average is taken over the things still present to be measured. When the failures have been quietly removed — funds closed, companies delisted, branches of the family that died out — the remainder looks better than the process ever was.

This compounds with the previous point in an unpleasant way. Multiplicative processes produce heavily skewed outcomes, so the survivors are few and enormous, and an average over them tells you almost nothing about what to expect. The median is the honest summary of a skewed population; the mean is a summary of its outliers.

Evidence multiplies too

Work in odds, not probability

Probability is a bad ruler for belief, because it is bounded at both ends and the space near the edges is enormously compressed. The step from 50% to 90% and the step from 90% to 99% feel utterly different and are, in the only sense that matters, the same size.

Odds fix this. 50% is 1:1, 90% is 9:1, 99% is 99:1. And Bayes multiplies odds:

Which means — take logs — that evidence adds. Alan Turing's team measured it in decibans for exactly this reason: it turned an accumulation of clues into a running total you could keep on a sheet of paper.

Press +10 dB three times from an even start. You go 50% → 91% → 99% → 99.9%. Three identical pieces of evidence, three identical steps along the rail, and three wildly different looking jumps in probability. The rail is telling the truth; the percentages are what is distorting.

A fact is only evidence if it discriminates

The likelihood ratio is P(evidence | hypothesis) ÷ P(evidence | not hypothesis). Both halves matter, and the second is the one people drop.

"Ninety per cent of burglars wear gloves" sounds damning until you ask what share of non-burglars wear gloves in January. If the answer is 85%, the likelihood ratio is 1.06 — +0.25 dB — and you have learned essentially nothing. Meanwhile a detail found in 30% of burglars and 1% of everyone else is a ratio of 30, or +14.8 dB, and it is worth more than fifty of the first kind.

The prior is a term in the sum

Because everything is addition now, the base rate is simply the number you start from — as real as any piece of evidence and often larger than all of them together.

A test that catches 95% of cases and false-alarms on 5% is worth 10·log₁₀(19) = +12.8 dB. That is a strong test. But if the condition affects three people in a thousand, you started at −25.2 dB, and a positive result leaves you at −12.4 dB, which is a probability of about 5%. The test is good. The prior is bigger.

Count each source once

Evidence only adds in full when it is independent. Two accounts of the same event are not two facts, and a second report from the same witness is worth a fraction of the first. The failure is the same one as holding two assets that move together and calling it diversification: in both cases you have counted one thing twice and feel better than you should.

Say what you actually believe

A scoring rule is proper if your best strategy under it is to report your true belief. The log score — you are scored on the logarithm of the probability you assigned to whatever actually happened — has this property, and it punishes overconfidence savagely while treating hedging gently.

Claim 99% and be wrong and you take 10·log₁₀(0.01) = −20 dB. Claim 60% and be wrong and you take −2.2 dB. The asymmetry is not a stylistic preference for caution. It is what makes stating your real credence the winning move.

The whole thing on one card

• A log is a change of ruler: equal distances become equal ratios.
• On that ruler multiplication is addition, and ×2 = +3 dB, ×10 = +10 dB.
• Big numbers are a few doublings. Steady growth has a doubling time of 70/r%.
• Losses and gains only cancel as ratios: −50% needs +100%.
• The average you are quoted is not the rate you grow at. The gap is ≈ σ²/2.
• So spread is a cost, and reducing it at a fixed average is free growth.
• Expected value describes a crowd. Growth describes you. Size bets by the second.
• Diversification raises growth; it is not a trade against return.
• Averages are taken over survivors. On skewed outcomes, prefer the median.
• Belief multiplies too, so evidence adds — in decibans, on top of the base rate.
• Evidence is a ratio of two likelihoods. The denominator is the one people forget.