Logarithms

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John Napier

John Napier (1550-1617), born near Edinburgh, Scotland, was Laird of Merchiston and a Scottish mathematician. He developed methods of rapid calculation, which he sought to apply to astronomy, trigonometry, navigation, mapmaking, and surveying. Napier discovered how to multiply numbers by doing the easier task of adding other, corresponding numbers that he called logarithms. Logarithms are used to describe natural phenomena mathematically. Napier also invented a set of "rods" or "bones" that could be arranged for arithmetical calculations.

Logarithms

Logarithms are numbers that are known in algebra as exponents or repeated multiplications of a single number. For example in the equation 3² = 27, 3 is the exponent and 2 is the base. Stated in terms of logarithms, 2 is the logarithm of the number 27 to the base 3. This statement can be written as log₃8 = 3. In general, if b^x=p, then x=log_bp .

The Properties of Logarithms

Because logarithms are exponents, the properties of exponents apply to them. The following equations show some of the important properties of exponents:

(1) b^x × b^y = b^{x + y}.

(2) b^x ÷ b^y = b^{x - y}.

(3) (b^x)^y = b^xy.

These properties of exponents can be restated as properties of logarithms: