LOGARITHM (from Gr. λόγος, word, ratio, and ἀριθμός, number), in mathematics, a word invented by John Napier to denote a particular class of function discovered by him, and which may be defined as follows: if a, x, m are any three quantities satisfying the equation ax = m, then a is called the base, and x is said to be the logarithm of m to the base a. This relation between x, a, m, may be expressed also by the equation x = loga m.
Properties.—The principal properties of logarithms are given by the equations
| loga (mn) = loga m + loga n, | loga (m/n) = loga m − loga n, |
| loga mr = r loga m, | loga r√ m = (1/r) loga m, |
which may be readily deduced from the definition of a logarithm. It follows from these equations that the logarithm of the product of any number of quantities is equal to the sum of the logarithms of the quantities, that the logarithm of the quotient of two quantities is equal to the logarithm of the numerator diminished by the logarithm of the denominator, that the logarithm of the r th power of a quantity is equal to r times the logarithm of the quantity, and that the logarithm of the rth root of a quantity is equal to (1/r)th of the logarithm of the quantity.
Logarithms were originally invented for the sake of abbreviating arithmetical calculations, as by their means the operations of multiplication and division may be replaced by those of addition and subtraction, and the operations of raising to powers and extraction of roots by those of multiplication and division. For the purpose of thus simplifying the operations of arithmetic, the base is taken to be 10, and use is made of tables of logarithms in which the values of x, the logarithm, corresponding to values of m, the number, are tabulated. The logarithm is also a function of frequent occurrence in analysis, being regarded as a known and recognized function like sin x or tan x; but in mathematical investigations the base generally employed is not 10, but a certain quantity usually denoted by the letter e, of value 2.71828 18284....
Thus in arithmetical calculations if the base is not expressed it is understood to be 10, so that log m denotes log10 m; but in analytical formulae it is understood to be e.
The logarithms to base 10 of the first twelve numbers to 7 places of decimals are
| log 1 = 0.0000000 | log 5 = 0.6989700 | log 9 = 0.9542425 |
| log 2 = 0.3010300 | log 6 = 0.7781513 | log 10 = 1.0000000 |
| log 3 = 0.4771213 | log 7 = 0.8450980 | log 11 = 1.0413927 |
| log 4 = 0.6020600 | log 8 = 0.9030900 | log 12 = 1.0791812 |
The meaning of these results is that
| 1 = 100, | 2 = 100.3010300, | 3 = 100.4771213, ... |
| 10 = 101, | 11 = 101.0413927, | 12 = 101.0791812. |
The integral part of a logarithm is called the index or characteristic, and the fractional part the mantissa. When the base is 10, the logarithms of all numbers in which the digits are the same, no matter where the decimal point may be, have the same mantissa; thus, for example,
In the case of fractional numbers (i.e. numbers in which the integral part is 0) the mantissa is still kept positive, so that, for example,
the minus sign being usually written over the characteristic, and not before it, to indicate that the characteristic only, and not the whole expression, is negative; thus
The fact that when the base is 10 the mantissa of the logarithm is independent of the position of the decimal point in the number affords the chief reason for the choice of 10 as base. The explanation of this property of the base 10 is evident, for a change in the position of the decimal points amounts to multiplication or division by some power of 10, and this corresponds to the addition or subtraction of some integer in the case of the logarithm, the mantissa therefore remaining intact. It should be mentioned that in most tables of trigonometrical functions, the number 10 is added to all the logarithms in the table in order to avoid the use of negative characteristics, so that the characteristic 9 denotes in reality 1, 8 denotes 2, 10 denotes 0, &c. Logarithms thus increased are frequently referred to for the sake of distinction as tabular logarithms, so that the tabular logarithm = the true logarithm + 10.
In tables of logarithms of numbers to base 10 the mantissa only is in general tabulated, as the characteristic of the logarithm of a number can always be written down at sight, the rule being that, if the number is greater than unity, the characteristic is less by unity than the number of digits in the integral portion of it, and that if the number is less than unity the characteristic is negative, and is greater by unity than the number of ciphers between the decimal point and the first significant figure.
It follows very simply from the definition of a logarithm that
The second of these relations is an important one, as it shows that from a table of logarithms to base a, the corresponding table of logarithms to base b may be deduced by multiplying all the logarithms in the former by the constant multiplier 1/loga b, which is called the modulus of the system whose base is b with respect to the system whose base is a.
The two systems of logarithms for which extensive tables have been calculated are the Napierian, or hyperbolic, or natural system, of which the base is e, and the Briggian, or decimal, or common system, of which the base is 10; and we see that the logarithms in the latter system may be deduced from those in the former by multiplication by the constant multiplier 1/loge 10, which is called the modulus of the common system of logarithms. The numerical value of this modulus is 0.43429 44819 03251 82765 11289 ..., and the value of its reciprocal, loge 10 (by multiplication by which Briggian logarithms may be converted into Napierian logarithms) is 2.30258 50929 94045 68401 79914 ....
The quantity denoted by e is the series,
| 1 + | 1 | + | 1 | + | 1 | + | 1 | + ... |
| 1 | 1·2 | 1·2·3 | 1·2·3·4 |
the numerical value of which is,
The logarithmic Function.—The mathematical function log x or loge x is one of the small group of transcendental functions, consisting only of the circular functions (direct and inverse) sin x, cos x, &c., arc sin x or sin−1 x,&c., log x and ex which are universally treated in analysis as known functions. The notation log x is generally employed in English and American works, but on the continent of Europe writers usually denote the function by lx or lg x. The logarithmic function is most naturally introduced into analysis by the equation
| . |
This equation defines log x for positive values of x; if x ≤ 0 the formula ceases to have any meaning. Thus log x is the integral function of 1/x, and it can be shown that log x is a genuinely new transcendent, not expressible in finite terms by means of functions such as algebraical or circular functions. A connexion with the circular functions, however, appears later when the definition of log x is extended to complex values of x.
A relation which is of historical interest connects the logarithmic function with the quadrature of the hyperbola, for, by considering the equation of the hyperbola in the form xy = const., it is evident that the area included between the arc of a hyperbola, its nearest asymptote, and two ordinates drawn parallel to the other asymptote from points on the first asymptote distant a and b from their point of intersection, is proportional to log b/a.
The following fundamental properties of log x are readily deducible from the definition
(i.) log xy = log x + log y.
(ii.) Limit of (xh − 1)/h = log x, when h is indefinitely diminished.
Either of these properties might be taken as itself the definition of log x.
There is no series for log x proceeding either by ascending or descending powers of x, but there is an expansion for log (1 + x), viz.
the series, however, is convergent for real values of x only when x lies between +1 and −1. Other formulae which are deducible from this
