Logarithms - An explanation of how they work

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.

Books of Log Tables were available for users.
Once you got the hang of doing Logs,
it was amazing how quickly you could calculate quite complex problems

They usually contained log base 10 tables as well as log base e,
usually refered to as Napierian or natural logs.
These books also gave tables for
Log. Sines, Cosines, Tangents, Cotangents, Secants, Cosecants,
and their Natural equivalents.

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