Page:Encyclopædia Britannica, Ninth Edition, v. 14.djvu/797

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

$\scriptstyle \log a\ b\times \log b\ a=1$,

$\scriptstyle \log b\ m=\log a\ m\times {\frac {1}{\log a\ b}}$.

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/log₁₀, which is called the modulus of the common system of logarithms. The numerical value of this modulus is 0.43429 44819 03251 82765 11289 &hellips;, and the value of its reciprocal, log, 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 + 1/1.2 + 1/1.2.3 + 1/1.2.3.4 + …

the numerical value of which is,

2.71828 18284 59045 23536 02874 …

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, arc cos x, &c., e^x, and log x, which are universally treated in analysis as known functions. It is the inverse of the exponential function e^x, the theory of which may be regarded as including that of the circular functions, since

$\scriptstyle \sin x={\frac {1}{2i}}\left(e^{ix}-e^{-ix}\right),$

$\scriptstyle \cos x=\left(e^{ix}+e^{-ix}\right)$.

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 − 1/2x² + 1/3x³ − 1/4x⁴ + &c.;

the series, however, is convergent for real values of x only when x lies between + 1 and − 1. Other formulæ which are deducible from this equation are given in the portion of this article relating to the calculation of logarithms.

We have also the fundamental formulæ:—

(i.)	Limit of $\scriptstyle {\frac {x^{h}-1}{h}}$, when h is indefinitely diminished, $\scriptstyle =\log x$;
(ii.)	$\scriptstyle \int {\frac {dx}{x}}=\log x+$ const.

Either of these results might be regarded as the definition of log x; they may be readily connected with one another, for we have in general

$\int x^{n}dx={\frac {x^{n}+1}{n+1}}+$ const.

but if n = −1, this formula no longer gives a result. Putting, however, n = −1 + h, where h is indefinitely small, we have

$\int {\frac {dx}{x}}={\frac {x^{h}-1}{h}}+$ const. = log x + const. by (i.).

The result (ii.) establishes a relation, which is of historical interest, between the logarithmic function and the quadrature of the hyperbola, for, by considering the equation of the hyperbola in the form xy = const., we see at once 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 pro portional to log b/a

The function log x is not a uniform function, that is to say, if x denotes a complex variable of the form a + ib, and if complex quantities are represented in the usual manner by points in a plane, then it does not follow that if x describes a closed curve log x also describes a closed curve; in fact we have

log (a + ib) = log √(a² + b²) + i(α + 2nπ),

where α is a determinate angle, and n denotes any integer. Thus, even when the argument is real, log x has an infinite number of values; for, putting b = 0 and taking a positive, in which case α = 0, we obtain for log a the infinite system of values log a + 2nπi. It follows from this property of the function that we cannot have for log x a series which shall be convergent for all values of x, as is the case with sin x, cos x, and e^x; as such a series could only represent a uniform function, and in fact the equation

log (1 + x) = x − 1/2x² + 1/3x³ − 1/4x⁴ + &c.,

is true only when the analytical modulus of x is less than unity.

The notation log x is generally employed in English works, but Continental writers usually denote the function by lx or lgx.

History.—The invention of logarithms has been accorded to [[Encyclopædia Britannica, Ninth Edition/John Napier|]], baron of Merchiston, in Scotland, with a unanimity which is rare with regard to important scientific discoveries. The first announcement was made in Napier’s Mirifici logarithmorum canonis descriptio (Edinburgh, 1614), which contains an account of the nature of logarithms, and a table giving natural sines and their logarithms for every minute of the quadrant to seven or eight figures. These logarithms are not what would now be called Napierian or hyperbolic logarithms (i.e., logarithms to the base e), though closely connected with them, the relation between the two being

e¹ = 10⁷e^−L/107, or L = 10⁷log_e10⁷ − 10⁷l,

where l denotes the logarithm to base e and L denotes Napier’s logarithm. The relation between N (a sine) and L its Napierian logarithm is therefore

N = 10⁷e^−L/107,

and the logarithms decrease as the sines increase. Napier died in 1617, and his posthumous work Mirifici logarithmorum canonis constructio, explaining the mode of construction of the table, appeared in 1619, edited by his son.

Henry Briggs, then professor of geometry at Gresham College, London, and afterwards Savilian professor of geometry at Oxford, admired the Canon mirificus so much that he resolved to visit Napier. In a letter to Ussher he writes, “Naper, lord of Markinston, hath set my head and hands at work with his new and admirable logarithms. I hope to see him this summer, if it please God; for I never saw a book which pleased me better, and made me more wonder.” Briggs accordingly visited Napier in 1615, and stayed with him a whole month. He brought with him some calculations he had made, and suggested to Napier the advantages that would result from the choice of 10 as a base, having explained them previously in his lectures at Gresham College, and written to Napier on the subject. Napier said that he had already thought of the change, and pointed out a slight improvement, viz., that the characteristics of numbers greater than unity should be positive and not negative, as suggested by Briggs. In 1616 Briggs again visited Napier and showed him the work he had accomplished, and, he says, he would gladly have paid him a third visit in 1617 had Napier’s life been spared.

Briggs’s Logarithmorum chilias prima was published, probably privately, in 1617, after Napier’s death, as in the short preface he states that why his logarithms are different from those introduced by Napier “ sperandum, ejus librum posthumus abunde nobis propediem satisfacturum.” The liber posthumus was the Canonis constructio already mentioned. This work of Briggs’s, which contains the