A logarithm is the **exponent**
given to a reference number (called the **base**)
in order to write a number in ‘shorthand’. For example, if we write 10^{2} =
100 where 2 is the exponent and 10 is the base, we say that log 100 = 2
wirh respect to base 10.
Two European mathematicians, **
John Napier** and **Henry Briggs**
worked out that logarithms could be used to simplify calculations in
physics and astronomy. There is good evidence that tables of the
logarithms of sines and cosines were in use by the Arab astronomer,
**Ibn Jounis** as early as the 13th
century.
Napier had been working on his invention of logarithms
for twenty years before he published his results, and this would place the
origin
of his ideas at about 1594. He had been thinking of the sequences which
had been published now and then of successive powers of a given number. In
such sequences it was obvious that sums and differences of indices of the
powers corresponded to products and quotients of the powers themselves;
but a sequence of integral powers of a base, such as 2, could not be used
for computations because the large gaps between successive terms made
interpolation too inaccurate. So to keep the terms of a geometric
progression of INTEGRAL powers of a given number close together it was
necessary to take as the given number something quite close to 1.
Napier therefore chose to use 1 - 10^(-7) or 0.9999999 as his given
number. To achieve a balance and to avoid decimals, Napier multiplied each
power by 10^7. That is, if N = 10^7[1 - 1/10^7]^L, then L is Napier's
logarithm of the number N. Thus his logarithm of 10^7 is 0. At first
he called his power indices "artificial numbers", but later he made up the
compound of the two Greek wordLogos (ratio) and arithmos (number).Napier
did not think of a base for his system, but nevertheless his tables were
compiled through repeated multiplications, equivalent to powers of
0.9999999 Obviously the number decreases as the index or
logarithm increases. This is to been expected because he was
essentially using a base which is less than 1. |