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JOHN NAPIER

( 1550 - 1617 )

PHILOSOPHER and MATHEMATICIAN

INVENTOR of LOGARITHMS ,  INVENTOR of the DECIMAL POINT


LOGARITHMS

An Explanation


Defined as:-  The logarithm of a number to a given base ( usually 10 or e ) is the index of the power to which the base must be raised to produce the number.

The whole number proceeding decimal point of the logarithm is known as the Characteristic,

the decimal fraction being called the mantissa. 

The characteristic of any number is found by subtracting 1 from the number of digits of the whole number,

ie; the numbers proceeding the decimal point.

By using a set of Logarithmic Tables,
the average school pupil could easily calculate complex numerical problems quite quickly,
once they had mastered  their Logarithms !!


Consider the following problem, 7777 multiplied by 218.

So, for a number of 777.70, the characteristic would be 2,

and 3 for our number of 7777.

The mantissa is found by referring to the portion of logarithm tables shown below.

So, locate the row for 77 and then locate column 7, this gives 0.8904.

But what about the value of the remaining digit 7, 

well that can be found looking along the same row to where there is a
group of columns headed ADD numbered 1 to 9.

Logarithms Base.10 A D D
log 0 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9
70 0.8451 0.8457 0.8463 0.8470 0.8476 0.8482 0.8488 0.8494 0.8500 0.8506 1 1 2 2 3 4 4 5 5
71 0.8513 0.8519 0.8525 0.8531 0.8537 0.8543 0.8549 0.8555 0.8561 0.8567 1 1 2 2 3 4 4 5 5
72 0.8573 0.8579 0.8585 0.8591 0.8597 0.8603 0.8609 0.8615 0.8621 0.8627 1 1 2 2 3 4 4 5 5
73 0.8633 0.8639 0.8645 0.8651 0.8657 0.8663 0.8669 0.8675 0.8681 0.8686 1 1 2 2 3 4 4 5 5
74 0.8692 0.8698 0.8704 0.8710 0.8716 0.8722 0.8727 0.8733 0.8739 0.8745 1 1 2 2 3 4 4 5 5
75 0.8751 0.8756 0.8762 0.8768 0.8774 0.8779 0.8785 0.8791 0.8797 0.8802 1 1 2 2 3 4 4 5 5
76 0.8808 0.8814 0.8820 0.8825 0.8831 0.8837 0.8842 0.8848 0.8854 0.8859 1 1 2 2 3 4 4 5 5
77 0.8865 0.8871 0.8876 0.8882 0.8887 0.8893 0.8899 0.8904 0.8910 0.8815 1 1 2 2 3 4 4 5 5
78 0.8921 0.8927 0.8932 0.8938 0.8943 0.8949 0.8954 0.8960 0.8965 0.8971 1 1 2 2 3 4 4 5 5
79 0.8976 0.8982 0.8987 0.8993 0.8998 0.9004 0.9009 0.9015 0.9020 0.9025 1 1 2 2 3 4 4 5 5

Locate the column for seven and read off the value 4.

Add this to the previous 0.8904 to give a mantissa of 0.8908

Therefore, the number 7777 would produce a logarithm whose characteristic would be 3, 

and the mantissa would be 0.8908, 

giving a complete logarithm of 3.8908

Similarly, our other number 218 produces a logarithm of 2.3385.

To multiply our two numbers, all we have to do is add the logarithmic equivalents of our numbers together.
 

so 7777 = log 3.8908
and  218 =   log 2.3385
total    = log 6.2293

    

which is fine, but how does that relate to an actual number, you may well ask !

Well, you have to look up what are called anti-logarithmic tables, which enable you to convert your logarithmic result back to a common number.

So using the antilogs table, below, we see that the logarithm mantissa 0.2293 gives a value of 1695.

Antilogarithms A D D
0 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9
0.20 1585 1589 1592 1596 1600 1603 1607 1611 1614 1618 0 1 1 1 2 2 3 3 4
0.21 1622 1626 1629 1633 1637 1641 1644 1648 1652 1656 0 1 1 2 2 2 3 3 4
0.22 1660 1663 1667 1671 1675 1679 1683 1687 1690 1694 0 1 1 2 2 2 3 3 4
0.23 1698 1702 1706 1710 1714 1718 1722 1726 1730 1734 0 1 1 2 2 2 3 3 4
0.24 1738 1742 1746 1750 1754 1758 1762 1766 1770 1774 0 1 1 2 2 2 3 3 4
0.25 1778 1782 1786 1791 1795 1799 1803 1807 1877 1816 0 1 1 2 2 2 3 3 4
0.26 1820 1824 1828 1832 1837 1841 1845 1849 1854 1858 0 1 1 2 2 3 3 3 4
0.27 1862 1866 1871 1875 1879 1884 1888 1892 1897 1901 0 1 1 2 2 3 3 3 4
0.28 1905 1910 1914 1919 1923 1928 1932 1936 1941 1945 0 1 1 2 2 3 3 4 4
0.29 1950 1954 1959 1963 1968 1972 1977 1982 1986 1991 0 1 1 2 2 3 3 4 4


The characteristic value of our log number is 6,
which means that there will be 7 digits ( ie: 6 + 1 ) before the decimal point in our result.

So the antilogs table gave us a value of 1695,

so we can write that down as a final answer of

1,695,000

Please note that this example used 4 digit Logarithm tables to the base 10.

7 digit tables would give a more accurate result.

For division,

the logarithm equivalents are subtracted from each other.

For numbers of value less than 1, please refer to further instructions.
( to follow soon )

Napier's original tables were to the base e, commonly referred to as natural logarithms.

Back to John Napier 


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