loga b × logb a = 1, logb m = loga m × (1/loga b).
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,
2.71828 18284 59045 23536 02874 ....
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
| log x = ∫x1 | dt | , (x > 0). |
| t |
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.