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preface to fifteenth edition

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2.102 SECTION 2<br />

2.2 MATHEMATICAL TABLES<br />

2.2.1 Logarithms<br />

2.2.1.1 Properties and Uses<br />

Definition of Logarithm. The logarithm x of the number N <strong>to</strong> the base b is the exponent of the<br />

power <strong>to</strong> which b must be raised <strong>to</strong> give N. That is,<br />

log N x or bx<br />

b<br />

N<br />

The number N is positive and b may be any positive number except 1.<br />

Properties of Logarithms<br />

1. The logarithm of a product is equal <strong>to</strong> the sum of the logarithms of the fac<strong>to</strong>rs; thus,<br />

log M · N log M log N<br />

b b b<br />

2. The logarithm of a quotient is equal <strong>to</strong> the logarithm of the numera<strong>to</strong>r minus the logarithm of<br />

the denomina<strong>to</strong>r; thus,<br />

M<br />

logb logb M logb<br />

N<br />

N<br />

3. The logarithm of a power of a number is equal <strong>to</strong> the logarithm of the base multiplied by the<br />

exponent of the power; thus,<br />

log Mp<br />

b p · logb<br />

M<br />

4. The logarithm of a root of a number is equal <strong>to</strong> the logarithm of the number divided by the index<br />

of the root; thus<br />

Other properties of logarithms:<br />

q 1<br />

logb pM logb<br />

M<br />

q<br />

q p<br />

p<br />

logb b 1 logb pM logb<br />

M<br />

q<br />

log 1 0<br />

log N log N · log a <br />

b b a b<br />

N<br />

log b N<br />

log b (b ) N b N<br />

log a N<br />

loga<br />

b<br />

Systems of Logarithms. There are two common systems of logarithms in use: (1) the natural<br />

(Napierian or hyperbolic) system which uses the base e 2.71828 ...;(2) the common (Brigg-<br />

sian) system which uses the base 10.<br />

We shall use the abbreviation log N log10<br />

N in this section.<br />

Unless otherwise stated, tables of logarithms are always tables of common logarithms.<br />

Characteristic of a Common Logarithm of a Number. Every real positive number has a real<br />

common logarithm such that if a b, log a log b. Neither zero nor any negative number has a<br />

real logarithm.<br />

A common logarithm, in general, consists of an integer, which is called the characteristic, and<br />

a decimal (usually endless), which is called the mantissa. The characteristic of any number may be<br />

determined from the following rules:

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