log(a) b × log(b) a = 1, log(b) m = log(a) m × (1/log(a) 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/log(a)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/log(e)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, log^(e) 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 log(e)
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 e^(x) 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
_
/ x dt
| log x = ---, (x > 0).
_/ 1 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.
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 (x^(h)-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.
log (1 + x) = x - 1/2 x^2 + 1/3 x^3 - 1/4 x^4 + ...;
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
equation are given in the portion of this article relating to the
calculation of logarithms.
The function log x as x increases from 0 towards [oo] steadily
increases from -[oo] towards +[oo]. It has the important property that
it tends to infinity with x, but more slowly than any power of x, i.e.
that x^{-m} log x tends to zero as x tends to [oo] for every positive
value of m however small.
The _exponential function_, exp x, may be defined as the inverse of
the logarithm: thus x = exp y if y = log x. It is positive for all
values of y and increases steadily from 0 toward [oo] as y increases
from -[oo] towards +[oo]. As y tends towards [oo], exp y tends towards
[oo] more rapidly than any power of y.