Two, Ten, and the Awkward One

Last timeMultiplication Becomes Addition

Every logarithm is the same curve with a different ruler. Which base a paper means is a matter of field rather than mathematics, and converting between them is one division.

A logarithm needs a base, and three of them are in common use. The practical

question is which one a given page means, and the mathematical question is how

much it matters.

The mathematical question has a short answer: not much. All logarithms are the

same curve stretched vertically, so everything qualitative is identical across

bases and only the units differ.

FIG 1The same curve, measured with three rulers

base 2: how many doublingsbase e: the natural one, written lnbase 10: how many factors of ten

Three curves that are one curve. Each is a fixed multiple of the others, so they all cross the axis at the same place, they all keep the same order, and the ratio of any two heights on one curve is the same as on the others. The base two curve is the tallest because doublings are the smallest steps, so there are more of them to count; the base ten curve is the shortest for the matching reason. Choosing a base is choosing a unit for the answer, in exactly the way that choosing between metres and feet is.

Changing base

Converting is a single division, and the formula is worth deriving rather than

looking up, because the derivation is two lines and the formula is easy to

invert by accident.

FIG 2Changing from one base to another

: what we want: the exponent that turns b into x
: what we have: the same question answered in whatever base the calculator or library offers, usually e
: the conversion factor, which depends only on the two bases and not at all on x. This is the number that makes the whole thing a rescaling
: any base at all. The left side does not mention c, so whatever is used on the right has to cancel out, and it does

The derivation is the power rule from the previous lesson. Write x as b raised to the log of x in base b, take the log in base c of both sides, and the power rule brings the exponent down as a factor, leaving the formula. The direction to be careful about is which log goes on the bottom: the base you are converting to is on the bottom, which can be checked in two seconds by trying x equal to b, where the answer has to be one.

There is an immediate and useful corollary. Because the conversion is a

constant factor, any expression that divides one logarithm by another is

unaffected by the base entirely. That covers the formula from the previous

lesson for the number of steps, and it is why the code there could use the

natural log for questions about doublings and get the right answer anyway.

The lesson stops here

5 more paragraphs to go

You have read the opening. The rest of the argument, the problems that check whether it landed, and the lines worth keeping at the end all come with a plan.

The first lesson of every course in the library reads the whole way through, free, so you can see exactly what the rest of them are.

See the plan The contents

This is the reading half

Starting the course gives you your own copy of it. Every idea on every page has problems standing under it, marked with a reason rather than a tick, and any sentence you do not believe can be opened and argued with. None of that can happen on a page nobody owns.

The contents

Two, Ten, and the Awkward One

Changing base

5 more paragraphs to go

This is the reading half

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