Multiplication Becomes Addition
Last timeThe Question a Logarithm Answers
Three rules turn products into sums, quotients into differences and powers into multiples. All three are exponent rules wearing a disguise, and deriving them once is cheaper than remembering them.
Logarithms are useful out of all proportion to their definition, and the
reason is one property: they convert multiplication, which is hard, into
addition, which is not.
That property is not an extra fact bolted onto the definition. It falls
straight out of it, in about two lines, and this lesson does the derivation
once so that the three rules never have to be remembered as rules.
- the base, the same throughout. Every rule here requires both logs to be in the same base, which is the one condition people drop
- the quantities being combined. Both have to be positive, since otherwise their logs do not exist, and the rule would be relating things that are not there
- any number at all, including negative and fractional ones. The third rule does not require k to be a whole number, which is what makes it so widely useful
- an identity in the sense of the previous course: true for every admissible value of the letters, not an equation to be solved
Where the first rule comes from
Recall from the previous course that multiplying two powers of the same base
adds the exponents, because the copies of the base are just being gathered into
one pile. Recall from the previous lesson that a logarithm is an exponent.
Put those two sentences together and the product rule is forced.
| step | line | what it says | why it is allowed |
|---|---|---|---|
| 1 | 1 | let p be the log of x, and q be the log of y | a definition, naming the two exponents we care about |
| 2 | 2 | then b to the p is x, and b to the q is y | line one rewritten as powers, which is the double arrow of the previous lesson |
| 3 | 3 | so x times y equals b to the p times b to the q | substitution: replacing each quantity by what it equals |
| 4 | 4 | which is b to the p plus q | the exponent rule for multiplication, which is where all of this comes from |
| 5 | 5 | so the log of x times y is p plus q | line four read back as a log statement, using the double arrow again |
| 6 | 6 | which is the log of x plus the log of y | substituting back the names from line one |
The third rule is the one that matters
The first two are conveniences. The third is a tool, because it is the only
thing in mathematics that reaches an exponent.
Consider the question the first lesson could only answer by counting: a
quantity multiplies by each step, and we want the number of steps until
it has grown by a factor . The equation is , and is
unreachable. It cannot be divided out, subtracted off or rooted away, because
it is not a thing being multiplied; it is the count of how many times.
Take the log of both sides. The power rule brings down in front, where it
is an ordinary factor:
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