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 log. 4.59859. Given log. 59859 nearest less log. 59857 Next less 59857 nat. numb. 3968 next greater 59868 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. Examples. Log. No. Log. No. 4.97436 = 94267 2.50181 = 317.55 2.97436 = 942.67 4.81584 = 65440 4.89538 = 78592 5•37081 = 234861 PROPOSITION VII. (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 sum. Required the product of 84 and 56. Logarithm of 84 1.92428 Logarithm of 56 1•74819 Product 4704 Log. = 3.67247 Required the product of 76.5 by 5.5. Logarithm of 76.5 1.88366 0:74036 Product 420•75 Log. = 2.62402 • Let 100 = a; 10~*=b; 10 -Y = c; 10% = d, &e. Then 10°—-—Y+% a b c d, &c. Where v is the logarithm of a; x the logarithm of b; the logarithm of 6; % the logarithm of d, &c. And v---y+% is the logarithm of a b c d. (A. I. and C. 2.) Required the product of •84 x 056 x 37. of .056 = 2:74819 + 3.24067 Sum of the decimals and affirmative index - 3. 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, subtract it from the index of that logarithm if it be negative, otherwise add it, before the signs are changed. Divide 3450 by 23. Logarithm of 3450 3:53782 Logarithm of 23 1.36173 1 (N) To involve a number to any power ; that is to square, cube, and 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 bers 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 num 3 The cube is 1.728 Log.=0.23754 Required the square of 25. Logarithm of 25 1.39794 2 The square is 625 Log. = 2.79588 a", where x * Let 10% = a, and n = the index of the power, then 10NX = a", Required the third power of .0725. The logarithm of .0725 is - 2.86034 and 2 X 3 = 6 product of the indices. Power •000381 log. = – 4:58102 Required the 6.25 power of .0032. the dec. and 3 x 6.25 = 18.75 Şproduct of the ind. Power 00000000000000025538log. - = 16:40719 PROPOSITION X. (0) To extract the square or cube root, 8c. of any number.* Divide the logarithm of the number by 2 for the squareroot, 3 for the cube-root, &c. and the quotient will be the 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 0.25527, the number answering to which is 1.8. What is the cube-root of 10648 ? Logarithm of 10648 is 4.02726, 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+0.58104; and by adding 2 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 index may divide even by •72) it becomes – 2.16 + 1.14227, 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 directions given in the 10th Proposition: for, the numerator denotes the power to which the given number is to be raised, and the denominator shows what root of that power is to be extracted. What is the value of .096 iš? 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 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. 1088303. (2.) Required the logarithm of 563' or •561. Ans. — 1•75076. 48 |