...and I, 2 &c. if it be only a decimal with 3, 4, &c. places. 6. The logarithm of a product consisting of any number of factors, is equal to the sum of the logarithms of those factors. Let a be the base : .r, x', x" &c. the logarithms of y, y', y" &c , then we have (by Def. 2). , „...

...properties of the indices of the same base. Hence, (1) In any system, the logarithm of a product, consisting of any number of factors, is equal to the sum of the logarithms of those factors. For, if a be the base of the system, and x, x', x", &c , the logarithms of n, n', n", &c., we have,...

...quantities is equal to the sum of the logarithms. And in general, the logarithm of a number consisting of any number of factors is equal to the sum of the logarithms of all its factors. (217.) It also follows from the above, that n times the logarithm of any number is...

...principle, which is merely an extension of that already proved in the preceding Article. The nth root of the product of any number of factors is equal to the product of the nth roots of those factors. Or in algebraic language For raise each of these expressions...

...radicals of the second, third, or n rt degree, according to the index of the root required. The n a root of the product of any number of factors is equal to the product of the n rt roots of the different factors. Or, in algebraic language, = V« XV * X liaise...

...principle, which is merely an extension of that already proved in the preceding Article. The nth root of the product of any number of factors is equal to the product of the nth roots of those factors. Or in algebraic language For raise each of these expressions...

...definition, r+x' is the logarithm of NN' ; that is to say, The logaritJim of the pro duct of two or more factors is equal to the sum of the logarithms of those factors. II. Divide equation (1) by (2). .•. by definition, x — z7 is the logarithm of ^ ; that is to say,...

...quantities themselves are equal : hence, y abc . . . . = ya X y SX ус . . . that is, íAe nth root of the product of any number of factors, is equal to the product of their nth roots. 1. Let us apply the above principle in reducing to its sim plest form the...

...the equation ; hence the logarithm of a negative number is imaginary. Properties of Logarithms. (1) The logarithm of the product of any number of factors...equal to the sum of the logarithms of those factors. Let p, Q, к, s be any quantities, and x, у, z, v their logarithms to any base a ; then if = P, a'...

...square root of 365, we decompose 365 into the prime factors 5 and 73, Now, it is obvious that the root of the product of any number of factors is equal to the product of their roots, hence, fhe square root of 365 is equal to the square root of 5 x 73, which...