RULE. Find the nearest less logarithm in the tables to the given one, and take out the four figures answering to it as before. Take the difference between this logarithm and the next greater in the tables, and also between this logarithm and the given one; divide the latter difference, with ciphers annexed to it, by the former, and place the quotient to the right hand of the natural number already found. Required the natural number answering to the logarithm 4.59859. Given log. 59859 nearest less log. 59857 next greater 59868 Next less 59857 nat. numb. 3968 diff. 2 annex ciphers. diff. 11 then 200 &c. 11 18 &c. the natural number is therefore 39681 8, the index being 4, there are five whole numbers. (L) To find the product of two whole or mixed numbers.* RULE. Add the logarithms of the numbers together, the natural number answering to the sum will be the product required. When several numbers are to be multiplied together, some of which are less than an unit, add the logarithms of the numbers together; when you come to the indices, add the affirmative indices and what you carry into one sum, and the negative indices into another. Take the difference between these sums for the index of the product, prefixing the sign of the greater the logarithm of c; z the logarithm of d, &c. And, v−a−y+z is the logarithm of a bed. (A. 1. and C. 2.) c; 10 = Required Required the product of 84 × 056 × 37. Logarithm of the product = 0·24067 The number answering to which is 1.7405, the product required. Required the product of 37 x 426 x 5 x 004 x 275 x 336. Answer 29.128. PROPOSITION VIII. (M) To divide one number by another.* Subtract the logarithm of the divisor from the logarithm of the dividend, and the remainder will be the logarithm of the quotient. If any of the indices be negative, or if the divisor be greater than the dividend, change the index of the divisor: then if the indices have unlike signs take their difference, and prefix the sign of the greater; if they have like signs take their sum, and prefix the common sign. When there is an unit to carry from the decimal part of the logarithm of the divisor, substract it from the index of that logarithm if it be negative, otherwise add it, before the signs are changed. * Let 10 = a, and 109 =c, then x is the logarithm of a, and y is the loga (N) To involve a number to any power; that is to square, cubes a number, &c.* RULE. Multiply the logarithm of the number by the number expressing the power, viz. by 2 for the square, 3 for the cube, 4 for the biquadrate, &c.; and the product will be the logarithm of the power required. If the index to the given logarithm be negative; multiply this index and the decimal part of the logarithm separately by the index of the given power, and subtract the former product from the latter; when you come to the indices, or whole numbers of these products, their difference must be taken, but if there be an unit to carry from the decimal part of the lower product, it must be added to the index of that product before you take the difference. The result will in every case be negative. Required the cube of 1.2 Logarithm of 1.2 0.07918 3 The cube is 1.728 Log. =0.23754 Required the square of 25. 1.39794 2 The square is 625 Log. is 625 Log. 2·79588 * Let 10" = a, ánd n = the index of the power, the 10a", where x is the logarithm of a, and n & the logarithm of a”. Required Required the third power of 0725. The logarithm of 0725 is -2.86034 Then 86034 x 3 = 2.58102 product of the decimal. and 2 x 3 = 6' Power '000381 log.-4.58102 Required the 6.25 power of 0032. product of the indices. The logarithm of 0032 is 3-50515. 3.15719 pro.of the de pro.of the ind and 3 × 6.25 = 18.75 Power 00000000000000025538 log.-16.40719 PROPOSITION X. (0) Toextract the square or cube-root, &c. of any number. Divide the logarithm of the number by 2 for the square root, 3 for the cube-root, &c. and the quotient will be th logarithm of the root. If the index to the logarithm be negative and does not exactly contain the divisor, increase it by such a number as will make it exactly divisible, and increase the logarithm also by the same number before you begin to divide. What is the square-root of 3.24? Logarithm of 3.24 is 0.51054, which divided by 2 gives 25527, the number answering to which is 1.8. What is the cube-root of 10648? Logarithm of 10648 is 402726, which divided by 3 gives 1'34242, the number answering to which is 22. What is the cube-root of 0003811? The logarithm of 0003811 is -4.58104 = = -4.58104-4+0.581041 and by adding to each part, it is 6+2-58104, divide by 3, then -2.86034 is the logarithm of the root, the number answering to which is 0725, the root sought. What is the '72 root of '096? The logarithm of '096 is — 2·98227 = −2+0·98227, and by adding 16 to each part (in order that the negative indes may divide even by 72) it becomes 2.161.14227, divide by 72 then ·3∙ + 1·58648 = -2.58648 is the logarithm of the root; hence the root sought is '03859. PROPOSITION XI. (P) To find the value of a quantity having a vulgar fraction for its exponent. RULE. Multiply the logarithm of the given number by the numerator of the exponent, and divide the product by the denominator; the quotient will be the logarithm of the quantity required. The multiplication must be performed as directed in the 9th Proposition, and the division according to the direc tions given in the 10th Proposition: for, the numerator denotes the power to which the given number is to be raised, and the denominator shews what root of that power is to be extracted. What is the value of "096) 3 3 ? The logarithm of '096 −2·98227 which multiplied by 25 (Prop. ix.) produces -26.55675; this divided by 18 (Prop. x.) gives 2-58648; the number answering to which is 03859. Answer. PROPOSITION XII. (Q) To find a fourth proportional to three given numbers; or to work a question in the rule of three by logarithms. RULE. Add the logarithms of the second and third terms together, and from the sum subtract the logarithm of the first term, the remainder will be the logarithm of the fourth term. What is the fourth proportional to 75; 36; and '008? Logarithm of 36= 1.55630 Logarithm of 0083.90309 Sum= 1-45939 Logarithm of 75187506 subtract. - 1.58433 the number answering to which is 384 the fourth proportional required. For, 75 36 :: 008: 384. PROMISCUOUS EXAMPLES, EXERCISING ALL THE PROPOSITIONS. (1.) Find the logarithm of 36. Ans. -1.88303. 48 (2.) Required the logarithm of 563′ or ·563. Ans. — 1·75076. Ans. (3.) Required the logarithm of 06'28803'. Ans.-2-79751. (4.) Find the logarithm of 0084'97133′. |