THE EXPONENTIAL THEOREM. 195. It is required to expand a3 in a series ascending by the powers of x. Since a=1+a-1; therefore a*={1+(a−1)}*, and by the Binomial Theorem we have {1+(a−1)}2=1+x(a−1)+2(x —1) (a−1)2+x(x—1) (x— 1.2 1.2.3 =1+{(a−1)—↓(a−1 )2 + } (a—1)3—¥(a—1)1+ . .... }x+Bx2 +Ca3... and if we put where B, C,.... denote the coefficients of x2, x3, Then a2=1+Ax+Bx2+Cx3+Dx1+Ex3+ ..... ax+h=1+A (x + h) + B (x + h)2 + C (x + h)3 + + Dat + Ah +2 Bah + 3Cx2h +4Dx3h +Bh2+3Cxh2 + 6Dx2h2 + + + + Ch3 + 4Dxh3 + + + Dh1 .) (1+Ah+Bh2+Ch3 + . . . .) =1+Ax+ B x2 + C 2-3 + D24 +.... +Bh2 + ABxh2 + B2x2h2 + + Ch3A Cxh3 + + Dh1 + Now these two expansions must be identical; and we must, therefore, have the coefficients of like powers of x and h equal; hence A = (a — 1) — + ( a − 1 )2 + ↓ (a − 1 )3 — † (a — 1 )1 + . . . . . Now, since this equation is true for every value of x, let x=1; then LOGARITHMS. 196. LOGARITHMS are artificial numbers, adapted to natural numbers, in order to facilitate numerical calculations; and we shall now proceed to explain the theory of these numbers, and illustrate the principles upon which their properties depend. DEFINITION. In a system of logarithms, all numbers are considered as the powers of some one number, arbitrarily assumed, which is called the BASE of the system, and the exponent of that power of the base which is equal to any given number is called the LOGARITHM of that number. Ꮖ Thus, if a be the base of a system of logarithms, N any number, and a such that N = a* then x is called the logarithm of N in the system whose base is a. The base of the common system of logarithms, (called from their inventor Briggs's Logarithms"), is the number 10. Hence since From this it appears, that in the common system the logarithms of every number between 1 and 10 is some number between 0 and 1, i. e. is a fraction. The logarithm of every number between 10 and 100 is some number between 1 and 2, i. e. is 1 plus a fraction. The logarithm of every number between 100 and 1000 is some number between 2 and 3, i. e. is 2 plus a fraction, and so on. 197. In the common tables the fractional part alone of the logarithm is registered and from what has been said above, the rule usually given for finding the characteristic, or, index, i. e. the integral part of the logarithm will be readily understood, viz. The index of the logarithm of any number greater than unity is equal to one less than the number of integral figures in the given number. "Thus, in searching for the logarithm of such a number as 2970, we find in the tables opposite to 2970 the number 4727564; but since 2970 is a number between 1000 and 10000, its logarithm must be some number between 3 and 4, i. e. must be 3 plus a fraction; the fractional part is the number 4727564, which we have found in the tables, affixing to this the index 3, and interposing a decimal point, we have 3.4727564, the logarithm of 2970. We must not, however, suppose that the number 3.4727564 is the exact log arithm of 2970, or that accurately. The above is only an approximate value of the logarithm of 2970 we can obtain the exact logarithm of very few numbers, but taking a sufficien number of decimals we can approach as nearly as we please to the true logarithm as will be seen when we come to treat of the construction of tables. 198. It has been shown that in Briggs' system the logarithm of 1 is 0, conse quently, if we wish to extend the application of logarithms to fractions, we mus establish a convention by which the logarithms of numbers less than I may be re presented by numbers less than zero, i. e. by negative numbers. Extending, therefore, the above principles to negative exponents, since It appears, then, from this convention, that the logarithm of every number between 1 and .1, is some number between 0 and 1; the logarithm of every number between .1 and .01, is some number between -1 and --2; the logarithm of every number between .01 and .001, is some number between - 2 and 3; and so on. From this will be understood the rule given in books of tables, for finding the characteristic or index of the logarithm of a decimal fraction, viz. The index of any decimal fraction is a negative number, equal to unity, added to the number of zeros immediately following the decimal point. Thus, in searching for a logarithm of the number such as .00462, we find in the tables opposite to 462 the number 6646420; but since .00462 is a number between .001 and .0001, its logarithm must be some number between 3 and -4, i. e. must be —3 plus a fraction, the fractional part is the number 6646420, which we have found in the tables, affixing to this the index -3, and interposing a decimal point, we have -3. 6646420, the logarithm of .00462. General Properties of Logarithms. 199. Let N and N' be any two numbers, x and x their respective logarithms, a the base of the system. Then, by definition, .. by definition, xa is the logarithm of N N', that is to say, The logarithm of the product of two or more factors is equal to the sum of the logarithms of those factors. The logarithm of a fraction, or of the quotient of two numbers, is equal to the logarithm of the numerator minus the logarithm of the denominator. III. Raise both members of equation (1) to the power of n. Nn = anx .. by definition, nx is the logarithm of N ", that is to say, The logarithm of any power of a given number is equal to the logarithm of the number multiplied by the exponent of the power. IV. Extract the nth root of both members of equation (1). The logarithm of any root of a given number is equal to the logarithm of the number divided by the index of the root. Combining the two last cases, we shall find, It is of the highest importance to the student to make himself familiar with the application of the above principles to algebraic calculations. The following examples will afford a useful exercise: We must not, however, suppose that the number 3.4727564 is the exact log arithm of 2970, or that 2970 = (10) 3.472756 accurately. The above is only an approximate value of the logarithm of 2970 we can obtain the exact logarithm of very few numbers, but taking a sufficient number of decimals we can approach as nearly as we please to the true logarithm, as will be seen when we come to treat of the construction of tables. 198. It has been shown that in Briggs' system the logarithm of 1 is 0, consequently, if we wish to extend the application of logarithms to fractions, we must establish a convention by which the logarithms of numbers less than I may be represented by numbers less than zero, i. e. by negative numbers. Extending, therefore, the above principles to negative exponents, since It appears, then, from this convention, that the logarithm of every number between 1 and .1, is some number between 0 and -1; the logarithm of every number between .1 and .01, is some number between 1 and --2; the logarithm of every number between .01 and .001, is some number between -2 and -3; and so on. From this will be understood the rule given in books of tables, for finding the characteristic or index of the logarithm of a decimal fraction, viz. The index of any decimal fraction is a negative number, equal to unity, added to the number of zeros immediately following the decimal point. Thus, in searching for a logarithm of the number such as .00462, we find in the tables opposite to 462 the number 6646420; but since .00462 is a number between .001 and .0001, its logarithm must be some number between - 3 and 4, i. e. must -3 plus a fraction, the fractional part is the number 6646420, which we have found in the tables, affixing to this the index 3, and interposing a decimal point, we have 3. 6646420, the logarithm of .00462. be General Properties of Logarithms. 199. Let N and N' be any two numbers, x and x their respective logarithms, a the base of the system. Then, by definition, |