# Logarithms

You can draw a graph of y=2x.

Then, for every value of y, look at the x-value needed to reach that y. Call it log2y.

log216 = 4 because 24 = 16

log28 = 3 because 23 = 8

log24 = 2 because 22 = 4

log21 = 0 because 20 = 1

log2½ = −1 because 2−1 = ½

There’s actually a difficulty we’re skating over in defining, say, log23, because it can be proved that no fractional power of 2 equals 3, and we don’t yet know what raising 2 to a power which can’t be written as a fraction means.

But skate over that, and we have a function log2x

And the index law 2x.2y = 2x+y

translates into the rule

log2AB = log2A + log2B

So logs “convert” multiplication sums into addition sums. And addition is easier than multiplication.

We just chose 2 for simplicity. We could choose any other positive number. The numbers most often used instead of 2 as what is called the “base” of logs are 10 and a special number, e, which turns out to be about 2.71828.

Logs were used as a practical aid to calculation for a long time before cheap and small calculators became available (1980s).

Every school student had either “log tables”

or a “slide rule”

Workers who did calculations as part of their job, and who didn’t work at a big desk calculating machine as many did, would also have a “slide rule” or log tables.

(Actually, many “computers” in the days before microelectronics – the word “computer” meant a person whose job was computing things, not a machine – were women. And engineers weren’t all white).

The “log tables” used log10, so for example log102=0.3010 because 100.3010 = 2 approximately.

You did a multiplication, say 230 × 34, by converting the numbers in the multiplication to “standard form”, 2.3×102 and 3.4×101

Then log 2.3 = 0.3617 from the tables
log 3.4 = 0.5315

Add the two values to get 0.8932

Look to see what number 0.8932 is the log of – roughly, log 7.82 = 0.8932