THE COMMON SYSTEM. 390. The base, in the common system of logarithms, is 10. Hence, since It thus appears that, in the common system, the logarithm of every number between 1 and 10 is some number between 0 and 1; that is, a proper fraction. The logarithm of every number between 10 and 100 is some number between 1 and 2; that is, 1 plus a fraction. The logarithm of every number between 100 and 1000 is some number between 2 and 3; that is, 2 plus a fraction; and so on. 391. By means of negative exponents the application of logarithms may be extended to numbers less than 1. Thus, From this it appears that the logarithm of every number between 1 and 0.1 is some number between 0 and 1; that is, 1 plus a fraction. The logarithm of every number between 0.1 and 0.01 is some number between - 1 and 2; that is, -2 plus a fraction. The logarithm of every number between 0.01 and 0.001 is some number between - 2 and 3; that is, 3 plus a fraction; and so on. 392. The logarithms of all numbers which are not exact powers of 10, being incommensurable with those numbers, can have their values only approximately obtained, and therefore are usually expressed by means of a decimal fraction. 393. The CHARACTERISTIC of a logarithm is its integral part. The decimal part is sometimes called the mantissa. 394. The characteristic of the logarithm of any INTEGER, or MIXED NUMBER, is positive, and equal to the number of integral places less one. This is evident from Art. 390. 395. The characteristic of the logarithm of any DECIMAL FRACTION is negative, and numerically one more than the number of ciphers between the decimal point and the first significant figure. This is evident from Art. 391. PROPERTIES OF LOGARITHMS. 396. The logarithm of 1 in any system is equal to 0. For, in the equation am, where x is the logarithm of m, whatever be the value of the base a, make x = 0, and we have a0 = m = 1, or log 1 = 0. 397. The logarithm of the base itself, in any system, is 1. For, make x= 1 in the equation aa=m, and we have a1 = m = a, or log a = 1. 398. A negative quantity has no real logarithm. For, since the base a is positive, all its powers must be positive (Art. 202). 399. The logarithm of a product is equal to the sum of the logarithms of its factors. For, let m = ď, and n = a3; then, multiplying these equations, member by member, we have Therefore, mn = aa" = a2+". log (mn) = x + y = log m+log n. 400. The logarithm of a quotient is equal to the logarithm of the dividend diminished by that of the divisor. For, let m = a, and n = a"; then, dividing the first equation by the second, member by member, we have 401. The logarithm of any power of a number is equal to the logarithm of the number multiplied by the exponent of the power. then, raising both members to the rth power, we have m2 = (a2)r = a*r. Therefore, log (m") =xrr log m. 402. The logarithm of any root of a number is equal to the logarithm of the number divided by the index of the root. then, extracting the rth root of both members, we have 403. The arithmetical mean of the logarithms of two numbers is the logarithm of the geometrical mean of those numbers. 404. From the foregoing, it is evident that multiplication, division, involution, and evolution may be performed by means of logarithms; hence they are often of great practical utility in shortening numerical calculations. In computations by logarithms, negative quantities are used as if they were positive (Art. 398), the sign of the result being determined irrespective of the logarithmic work. COMPUTATION OF LOGARITHMS. 405. The properties just demonstrated are true of every system of logarithms; but their application to numerical calculations supposes the construction of a logarithmic table. A table of logarithms may be computed by means of Art. 403; but the following is a more convenient method. Let us resume the equation a2 = n, where x is the logarithm of n, to the base a. (1) a=1+m, and n = 1+p, Assume then (1 + m)* = 1 + p. (2) Raising both members of this equation to the nth power, we have (1+m)”* = = (1+p)", for every value of n. Expanding both members of the last equation by the Binomial Formula, we obtain Omitting 1 from both members, dividing by n, and factoring the first member, we find Make n 0, and the last equation reduces to Hence, if we make the reciprocal of the other factor of the same member equal to M, equation (3) becomes which is an expression for the logarithm of any number n. This expression is composed of two factors; the one within the parenthesis, dependent upon the number, and the other, M, dependent upon the base of the system. The factor dependent upon the base is called the modulus of the system of logarithms. |