4. The logarithm of the product of two or more positive numbers is found by adding together the logarithms of the several factors. For, MX Nam X an = am+n. loga (M × N)=m+n=log。M+logɑN. Similarly for the product of three or more factors. 5. The logarithm of the quotient of two positive numbers is found by subtracting the logarithm of the divisor from the logarithm of the dividend. 6. The logarithm of a power of a positive number is found by multiplying the logarithm of the number by the exponent of the power. 7. The logarithm of the real positive value of a root of a positive number is found by dividing the logarithm of the number by the index of the root. Change of System. Logarithms to any base a may be converted into logarithms to any other base b as follows : Let N be any number, and let Then, n=loga N and m=log, N. Nan and N—¿m. ... an—fm Taking logarithms to any base whatever, n log a = m log b, or, log a X log, N= log b × log, N, from which log, N may be found when log a, log b, and log, N are given; and conversely, loga N may be found when log a, log b, and log, N are given. Two Important Systems. Although the number of different systems of logarithms is unlimited, there are but two systems which are in common use. These are: 1. The common system, also called the Briggs, denary, or decimal system, of which the base is 10. 2. The natural system of which the base is the fixed value which the sum of the series approaches as the number of terms is indefinitely increased. This fixed value, correct to seven places of decimals, is 2.7182818, and is denoted by the letter e. The common system is used in actual calculation; the natural system is used in the higher mathematics. EXERCISE XXIII. 1. Given log102=0.30103, log103=0.47712, log10 7-0.84510 find log106, log1014, log1021, log104, log1012, 3. Given log10 e 0.43429 find log 2, log 3, log. 5, log. 7, log. 8, 4. Find x from the equations 5=12, 16-10, 27*=4. This equation is true for all real values of x, since the binomial theorem may readily be extended to the case of incommensurable exponents (College Algebra, § 264); it is, however, only true for values of n numerically greater than must be numerically less than 1 (College Algebra, 1, since § 375). 1 N As (1) is true for all values of x, it is true when x=1. This last equation is true for all values of n numerically greater than 1. Taking the limits of the two members as n increases without limit we obtain and this is true for all values of x. It is easily seen that both series are convergent for all values of x. The sum of the infinite series in parenthesis is the natural base e. This equation is true for all values of n greater than x (College Algebra, § 375). Take the limit as n increases without limit, x remaining finite; then If n is merely a large number, but not infinite, where is a variable number which approaches the limit 0, € when n increases without limit. Hence 1 + 2/2 = √1 + x + e, n y=n√1+x+e¬n. |