degree of the power, and find, in the tables, the number corresponding to the product. Evolution by Logarithms. (112.) Let ay: then if we extract the 2d, 3d, 4th, and nth roots of both members of this equation, we shall have, (45,) 3 až vy, a3 =vy, aẪ =√y, or, 1.y = 1. Vy, † 1.y = 1. Vy, † ly 1 añ =Vy; = from which it follows, that the logarithm of any root of a number is the logarithm of the number itself divided by the index of the root. Reduction of Exponential Equations by Logarithms. (113.) If we take the general equation bc, we shall have, by supposing the logarithms of both members known, 1.6* = l.c. It has been shown, that the logarithm of any power of a number, is equal to the logarithm of the number itself, multiplied by the exponent of the power (111); hence, 1.b* = x l.b, and therefore we have, x l.b=l.c; For a numerical example; let it be required to find the value of x from the equation 8 128, we shall have, S*= (114.) Having shown how numerical calculations are performed by means of logarithms, we will now consider, more particularly, the properties of the common system of logarithms, formed upon the base 10. We have seen, that in this system, the log. of 1=0, 1.101, 1.1002, 1.10003, 1.10000=4, and that all intermediate numbers have for their logarithms, the same whole numbers, plus a fraction. The integral part of a logarithm can, therefore, be known from the number of places of figures contained in the number to which it belongs. For numbers less than 10, it is 0; for those greater than 10, and less than 100, it is 1; for those greater than 100, and less than 1000, it is 2, &c., being always one less than the number of figures expressing the given number. The integral part of a logarithm is called the characteristic of the logarithm. (115.) It may also be shown, that the decimal part of the logarithms of numbers, one of which is 10, 100, 1000, &c. times greater or less than the other, is the same; for the logarithms of 10, 100, 1000, &c. are whole numbers, and being added or subtracted, would only alter the value of the characteristic, and would not affect the decimal part. For example; 1.28 × 10, or 1.280 = 1.28 +1.10, 1.28 × 100, or 1.2800 1.28 × 1000, or 1.28000 = 1,447158 +1.2,447158, 1,447158+2.3,447158, 1,447158 +3.4,447158. = For the same reason, the logarithm of any number of figures, whether whole numbers or decimals, will be the same, with the exception of the characteristic, which will always be one less than the number of figures in the integral part of the number. For example; taking the number 654685, and dividing it successively by 10, and for each division subtracting from its logarithin the logarithm of 10, we have, From this example, it also appears, that the characteristic of a decimal fraction is negative. It is always determined by the place which the first significant figure of the decimal occupies, counting from the decimal point towards the right. If it occupy the first place, the characteristic is -1; if the second, it is -2, &c., being obtained by subtracting the logarithm of 10, 100, 1000, &c. from zero. (116.) The logarithms of all fractional numbers are negative, since they result from the subtraction of a greater loga rithm from a less. To obtain the logarithm of, for exam1.2 1.3 1. 3, 1., ple, we have (110), = or, 0,3010300-0,4771213 = 0,1760913. Negative logarithms would be inconvenient in calculation, and to avoid their use we reduce the fraction to a decimal, the logarithm of which is positive, except the characteristic. To effect this reduction, we have only to consider the numerator multiplied by 10, 100, or 1000, as may be necessary, which is done by adding 1, 2, or 3 to its characteristic, and then subtract the logarithm of the denominator. The number of units necessary to be added to the characteristic of the numerator, will determine the characteristic of the resulting logarithm, which must always, as has been shown (115), be negative. Thus, to obtain the logarithm of 3, we multiply the numerator by 10, which gives for the numerator, 20 tenth parts of a unit, which divided by 3, will give a quotient in tenth parts. To effect this division by logarithms, we subtract the log. of 3 from the log. of 20: and since this difference is the log. of 10th parts, we must affix to it - 1 for a characteristic; or, which is the same thing, considering this difference the log. of a whole number, subtract the log. of 10 from it, to effect the division by 10, which gives 1 for the characteristic. In practice, it is usual to perform the subtraction, as though the logarithm of the numerator were the greater, till we come to the characteristic, we then take the excess of the characteristic of the log. of the denominator over that of the numerator, and affect it with the negative sign, for the characteristic of the resulting logarithm. The result is the same as by the preceding method. For example; let it be required to find the quotient of 5, divided by 73 in decimals, by means of logarithms. nearly; but the characteristic, -2, shows that this number must be a decimal fraction, the first significant figure of which stands in the place of hundredths (115); we have, therefore, 0,068491 for the quotient of 5 divided by 73, extended to six places of decimals. (117.) We now proceed to show the application of logarithms to algebraic formulas. The student should be provided with a table of logarithms, and exercise himself in their use by giving numerical values to the algebraic quantities, and performing the operations indicated by means of logarithms. Let it be required to find the last term of a geometrical progression by means of logarithms; the first term, the ratio, and the number of terms being given. Calling a the first term, r the ratio, and n the number of terms, we have (94), l=ar"-1; taking the logarithms 1.l = 1.a+(n−1)×l.r (109), (121). If 1, r, and n be known, we find, a = logarithms l.al.l—(n − 1)1.r (110). And taking the If a, l, and n be given, and we would find r, we have, If a, l, and r be known, and we would find n, we have, a(r” — We might also take the formula S = a(r" — 1); r 1 and supposing three of the quantities which enter into it known. Find the other by means of logarithms. Thus, n, a, and → being known, we have, S= a(r" — 1) r And by logarithms 1.Sl.a +1.(r” — 1) — 1.(r — 1), and by logarithms, nl.rl.(rS — S+ a) — 1.a; 1.(rS-S+a)-1.a and, n = 1.r (113). Proportion by Logarithms. α (118.) Let there be the proportion a:bc:d; then we bc shall have d == and by logarithms, l.dl.bl.c— l.a; hence, to find the fourth term of a proportion, add together the logarithms of the 2d and 3d terms, and subtract from their sum the logarithm of the first term; the remainder will be the logarithm of the fourth term. FORMULAS RELATING TO INTEREST AND ANNUITIES. Simple Interest. (119.) Let the principal, or sum on which interest is to be calculateda, the rate per cent., or the fraction expressing the interest of $1 for one year = = r; the time for which the principal a is loaned = t; and the amount at the expiration of the time t A; then we shall have the proportion as $1 is to r, its interest for one year, so is the principal a, to its interest for one year, or, 1:ra: ar; ar is therefore the interest of a for one year. But if the principal is loaned for a time greater or less than one year, the interest will be proportional to the time which will give the proportion, 1 ; ar = art; art would therefore be the interest of a for the time t. Thus, if a = 230, r = =,06, and t = 2 years and 5 months, we shall have, art=230,06 × 2,533,35. If, the other values remaining the same, t = 67 days, then art=230 ×,06 × 0.75 2,533. 36 = t: (120.) If we would know the amount, we have only to add the principal to the interest, and we shall have A=a+art. From this last formula, others may be derived by bringing each of the quantities which enter into it, to stand alone on one side of the sign of equality; thus, |