The Question a Logarithm Answers
Last timeGrowing by Multiplying
A logarithm is the exponent you needed, and nothing else. Reading every log expression back as that question is the whole technique, and it removes the need to remember any rule.
The previous lesson left a question hanging. A quantity multiplies by a factor
each step, and we want to know how many steps until it reaches some amount.
Counting works, and counting does not scale: it cannot be put in a formula, it
cannot be solved for anything, and it gives no insight into how the answer
depends on the factor.
What is needed is the inverse question. Powers go one way: here is a base, here
is how many times to multiply it by itself, what do you get. The logarithm goes
the other way: here is a base, here is what we got, how many times was it
multiplied.
That is the entire definition. Everything else in this lesson is that sentence
applied in various places.
- the base: the number being multiplied by itself repeatedly. It has to be positive and not one, since one multiplied by itself is always one and the question would have no answer
- the exponent: how many times. This is the thing the logarithm hands back, so if you want to know what a log is, it is this
- what you got. It has to be positive, because no exponent of a positive base ever produces nought or a negative number
- the double arrow of the previous course: each side implies the other, so these are two spellings of one fact rather than two facts
Reading one aloud
The symbol is an operation in appearance only. It is better read as a phrase,
and the phrase is: the exponent that turns the base into this.
Once an expression is read that way it often answers itself. The log of a
thousand to base ten is the exponent that turns ten into a thousand, which is
three, and no calculator was involved. The log of thirty two to base two is the
exponent that turns two into thirty two, which is five, counted on fingers.
| step | as written | read aloud as a question | the answer, and why |
|---|---|---|---|
| 1 | log base 2 of 8 | what exponent turns 2 into 8 | 3, because 2 times 2 times 2 is 8 |
| 2 | log base 10 of 1000000 | what exponent turns 10 into a million | 6, because a million is 1 followed by six noughts |
| 3 | log base 2 of 1 | what exponent turns 2 into 1 | 0, because leaving the base alone is what a nought exponent means |
| 4 | log base 2 of 0.25 | what exponent turns 2 into a quarter | minus 2, because shrinking needs a negative exponent |
| 5 | log base 10 of 0 | what exponent turns 10 into nought | there is none, which is why this expression is undefined |
| 6 | log base 2 of 10 | what exponent turns 2 into 10 | between 3 and 4, since 2 cubed is 8 and 2 to the 4th is 16; about 3.32 |
A log appearing inside a larger formula is read exactly the same way. If a
paper writes the negative of the log of a probability, the reading is: take the
exponent that turns the base into that probability, which is negative because
probabilities are below one, and flip the sign so it is positive. Having read
it, you have a quantity, and quantities can be added, compared and minimised
without any further thought about where they came from.
The lesson stops here
4 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 planThe contentsThis 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