ChristopherLind.com -- The Homepage of Chris Lind With a deep passion for learning and knowledge, Chris is an avid enthusiast of mathematics, computer science, physics, philosophy of mind, and economics. If you have any questions or comments about what you read on this website, feel free to email him at: NoSpam@ChristopherLind.Com (Replace NoSpam with Chris) Article: Logarithms, the FIRST calculators 5-6-2005

## Logarithms, the FIRST calculators

The hand held calculator that everyone uses today first became available in the early 1970's. Prior to this time, if you wanted to do an arithmetic calculation, you had to do it by hand. Obviously, this takes ALOT of time, and so early on people began looking for ways to reduce the amount of work involved in multiplication and division.

In the late 16th century, a man named John Napier was one of those people searching for a way to make arithmetic calculations easier. At this time, the properties of powers(exponents) were known. Napier saw a PATTERN in the properties of the powers of numbers that led him to his discovery

What is a power?
Raising a number to a power simply means multiplying a number by itself.

For example,
22 reads "2 raised to the second power".
22 means 2 X 2 = 4

23 reads "2 raised to the third power".
23 means 2 X 2 X 2 = 8

2 is called the "base" and 3 is called the "exponent" or "power".

Here is a table of 2 raised to different powers.
 reads means 21 "2 raised to the first power" 2 22 "2 raised to the second power" 2 X 2 = 4 23 "2 raised to the third power" 2 X 2 X 2 = 8 24 "2 raised to the fourth power" 2 X 2 X 2 X 2 = 16 25 "2 raised to the fifth power" 2 X 2 X 2 X 2 X 2 = 32 26 "2 raised to the sixth power" 2 X 2 X 2 X 2 X 2 X 2 = 64 27 "2 raised to the seventh power" 2 X 2 X 2 X 2 X 2 X 2 X 2 = 128 28 "2 raised to the eighth power" 2 X 2 X 2 X 2 X 2 X 2 X 2 X 2 = 256

If you think about it, you will realize that
if 22 = 2 X 2
and 23 = 2 X 2 X 2
then 22 X 23 = (2 X 2) X (2 X 2 X 2).

In other words, 22 X 23 = 2(2 + 3) = 25

if you replace 2 with the letter A, then you will realize that
if A2 = A X A
and A3 = A X A X A
then A2 X A3 = (A X A) X (A X A X A).

In other words, A2 X A3 = A(2 + 3) = A5

From looking for patterns in exponents, Napier realized that if you could rewrite all of the numbers in terms of a common base (base A, for instance), then you could reduce multiplication to addition, division to subtraction, powers to multiplication, and roots to division....

Napier's "invention" was literally to create a table of exponents as above. Only in this table, the exponent itself would be a seperate column. Napier first wanted to name his table, the table of "artificial numbers", but later decided to call the table a table of "logarithms".

Here is a base 2 table of logarithms:
 Exponent 21 1 2 22 2 4 23 3 8 24 4 16 25 5 32 26 6 64 27 7 128 28 8 256 28 9 512 28 10 1024

So how does this table make multiplication easier?

Say you wanted to multiply the numbers 64 and 16 together.
Looking at the entry for 64, you see that the exponent is 6.
Looking at the entry for 16, you see that the exponent is 4.

6 + 4 = 10
Looking down the exponent column for 10, you find the number 1024, which is the ANSWER!

By using this table, multiplication is reduced to addition. Also, division is reduced to subtraction, powers are reduced to multiplication, and roots to division.

In this table, the exponent column is also called the "logarithm" of the number in the right most column.

In other words,
log2(64) = 6
log2(16) = 4
log2(1024) = 10