Logarithms

2x

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

04blog1

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”

log10_tables_4_figure

or a “slide rule”

figure1_multiplication_c-d

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.

150-extra-engineers-with-slide-rules

(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

Then your answer is 7.82×103 in standard form.

On the slide rule, you moved the slide so that the “1” on the slide was above 2.3 on the frame, and then moved the cursor so that the line in the cursor passed through 3.4 on the slide. Look down below 3.4 on the slide to the number on the frame, and it’s 7.82.

Calculators are much quicker and more accurate, and logs to base 10 are less used these days.

They are still used a bit in science. For example, a decibel, the unit of sound volume, is defined as ten times the logarithm to base 10 of the ratio of two power quantities.

The Richter scale measure of the size of an earthquake is defined from the log to base 10 of the amplitude of waves recorded by seismographs.

So a sound 10 decibels greater than another is 10 times as loud, and 20 decibels greater, 100 times as loud.

An earthquake one point bigger on the Richter scale than another produces ten-times-bigger waves on seismographs.