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Article: Logarithms, the FIRST calculators 562005
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,
2^{2} reads "2 raised to the second power".
2^{2} means 2 X 2 = 4
2^{3} reads "2 raised to the third power".
2^{3} 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 
2^{1}  "2 raised to the first power"  2 
2^{2}  "2 raised to the second power"  2 X 2 = 4 
2^{3}  "2 raised to the third power"  2 X 2 X 2 = 8 
2^{4}  "2 raised to the fourth power"  2 X 2 X 2 X 2 = 16 
2^{5}  "2 raised to the fifth power"  2 X 2 X 2 X 2 X 2 = 32 
2^{6}  "2 raised to the sixth power"  2 X 2 X 2 X 2 X 2 X 2 = 64 
2^{7}  "2 raised to the seventh power"  2 X 2 X 2 X 2 X 2 X 2 X 2 = 128 
2^{8}  "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 2^{2} = 2 X 2 and 2^{3} = 2 X 2 X 2
then 2^{2} X 2^{3} = (2 X 2) X (2 X 2 X 2).
In other words, 2^{2} X 2^{3} = 2^{(2 + 3)} = 2^{5}
if you replace 2 with the letter A, then you will realize that
if A^{2} = A X A and A^{3} = A X A X A
then A^{2} X A^{3} = (A X A) X (A X A X A).
In other words, A^{2} X A^{3} = A^{(2 + 3)} = A^{5}
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  
2^{1}  1  2 
2^{2}  2  4 
2^{3}  3  8 
2^{4}  4  16 
2^{5}  5  32 
2^{6}  6  64 
2^{7}  7  128 
2^{8}  8  256 
2^{8}  9  512 
2^{8}  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,
log_{2}(64) = 6
log_{2}(16) = 4
log_{2}(1024) = 10

