8

Logarithms and Trigonometrjcal
Ratios

The system of logarithms is based upon the fundamental fact
that every integer can be expressed as a power of any other
integer, the latter then being the base to which the loga-
rithms are calculated. If any two numbers to be multiplied
together are first converted to powers of the same base num-
ber, then the special rules affecting their exponents apply,
and the task of multiplying is converted to one of addition.
Ordinary logarithm tables in everyday use are calculated to
the base of 10.

The numbers 10, 100, 1000 and 10,000 form a geometric
progression as follows:

Progression: 1oL yot o vhe 10t
Exponent of term: 1 2 3 4
Progression: I0 I00 1000 10000

The two progressions are of course the same. The terms in the
third row are converted, in the first row, to powers of 10 and
the exponents of these power numbers are shown in the
middle row. In order to multiply 10 by 1000 (that is, the
first term by the third) we find the answer in the fourth (that
is, 1+3) column.

This is the skeleton framework of our logarithm tables.
The terms in the second row are the logarithms of the terms
in the bottom row and, conversely, those in the bottom row
are the antilogarithms of those in the second row. The com-

plete tables as now used were compiled by filling in the gaps
6 81

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