CHAPTER III LOGARITHMS 12. Definitions and preliminary notions. In elementary algebra the power a" is denned for positive, negative, zero, and fractional values of x. If a" = y, (.4) the number a is called a base, y is a number, and x is called the logarithm of y to the base a. The logarithm of a number to a given base is defined to be the. exponent which indicates the power to which the base must be raised to produce the given number. We write ^ = log,y> (-5) so that equations (^4) and (5) are only different ways of saying the same thing. It can be shown that when a> 0 and a^ 1, a single real posi- tive value of y exists for every real value of a-, even though x be an incommensurable number, like "v2; and, conversely, for every real positive value of y a single real value, which may be positive or negative, exists for x. In other words, given a positive base a, different from unity, corresponding to every real logarithm, there exists a single positive number; and, conversely, for every real positive number there exists a single real logarithm. Negative numbers have no real logarithms, for the base a is always taken as a positive number, and the negative numbers that might otherwise exist, as in an equation like 10"=y, have been deliberately excluded from the definition of the expression a". A set of logarithms x of all numbers y, corresponding to a given base a, is called a system of logarithms. Two systems are in use, the common, or Briggsian, system, which is used in practical computations, and the hyperbolic, or Napierian, system, which is always used in analytical work. The base of the common system is 10, while the base of the Napierian system is the number e = 2.71827+, which was found in § 10. 80 LOGARITHMS 31 The following theorems hold for all systems of logarithms: ', I. The logarithm of'1 is zero. For, suppose log,,! = x. This fact may be written a" == 1. But when a is different from unity, the only power of a that can be equal to 1 is a". _. ^y 1=0. II.The logarithm of the. base is 1. For, let log,, a =a;. Putting this statement in the form of equation (J), a^a, and a;=l is the only real value of x which will satisfy this equation. III.The logarithm of a product is equal to the sum of the loga- rithms of the factors. For, let x and y be the factors of a product and m and n their respective logarithms. Then ^go2' = m and ^SaS/ = w' or x = a"1 and y = a\ Consequently, xy=am+" and log,, xy = m + n = log,, x + log^y. f IV.The logarithm of a power of a number is equal to the loga- rithm of the number multiplied by the exponent of the power. Let x be the number, n the exponent of the power, and ; the logarithm of x. Then we have ;=log,,a-, or a1 =x; whence a-" = a"1 and, consequently, log^a;" = nl = n log^a". '' COROLLARY. The logarithm of a quotient is equal to the loga- rithm of the dividend diminished by the logarithm of the divisor. For the quotient - may be written xy~1. We have then y log,^ = log^a-y-1 = log^a- + log,y-1 = log^a- - log^y. y