which case you would subtract the quotient obtained by dividing the difference as above instead of adding it. EXAMPLE. Find the number the log. of which is 5.868789 The decimal part of the next less log. is that of 73925.868762 Their difference is 27 Annex ciphers to this diff. and divide by the number in column D which is 59. Annex the quotient 45 to the number 7392 before found, and you have the number required corresponding to the given logarithm, namely 739245. This number contains 6 figures, one more than the characteristic of the given logarithm. In every case a sufficient number of ciphers must be annexed to obtain quotient figures enough, when appended, to make the whole of the number which thus results contain one more figure at least than is expressed by the characteristic of the given logarithm. If more quotient figures still be obtained, they will occupy the place of decimals. log. of 26784* is 4.427875 Their sum is 5.323574 5.323574 is the log. of 210656, which last number is, therefore, the product required. EXAMPLE II. Required the product of 3.586, 2.1046,.8372 and .0294. log. of 3.586 is 0.554610 of 2.1046 is 0.323170 of .8372 1.922829 of 0294 2.468347 difference 7.352220 7.352220 is, therefore, the log. of the quotient which from the tables, observing the converse rule for pointing off decimals according to the characteristic, (4 art. 50) is .225019. * In looking for the log. of this number, look first for that of 2678, multiply the tab. dif. by 5, the last figure of the given number, and cut off one figure from the product. EXAMPLE IV. Divide .06314 by 007241. log. of .06341 of .007241 is 2.800305 = n Quotient 8.71979 0.940506 54. We shall now demonstrate rules for raising numbers to powers, and for extracting the roots of numbers, by means of logarithms. Resume the equation, a raising both members to the mth power, we have, observing the rule of Algebra, which is to multiply the exponent by the degree of the power. alm from this last equation, it appears that im is the power to which it is necessary to raise the base a in order to produce nm; hence the following = nm RULE. a To raise a number to any power, by means of logarithms, multiply the logarithm of the given number by the exponent of the power, and the product will be the logarithm of the power. EXAMPLE I, Required the 4th power of .09163 log. of .09163 is 2.962038 Multiply by 4 Product 5.848152 5.849152 is the log. of .0000704939, which last number is the power required. EXAMPLE II. Required the tenth power of .64. log of .64 1.806180 10 Power .0115292 2.061800 In multiplying the first decimal place by 10, the product is 80, then 8 times ī is 10, and 8 to carry is 2. 55. To find a rule for extracting the root of a number by means of logarithms assume again the equation. al = n Take the mth root of both members applying in the first member the rule to divide the exponent by the number expressing the degree of the root, and there results. m am —Vn 1 is here plainly the logarithm of vn; hence the following m RULE. To extract the root of a number by means of logarithms, divide the logarithm of the given number, by the index of the root, and the quotient will be the logarithm of the root. EXAMPLE I. Required the 4th root of 434296. log. of .434296 1.637786 $ of this logarithm is obtained by observing that the index which alone is negative, must be divided separately, as we should divide a minus term, followed by a plus term in Algebra ; the I can be rendered divisible by borrowing 3, and afterwards carrying +3 before the 6, rendering it 36 ; that is, the proposed logarithm is viewed under the form 4+3.637786. The quotient is 1.909446,which is the logarithm of.811795, the fourth root required. EXAMPLE II. . Required the 10th root of 2. log. of 2 0.301030 Divide this by 10 0.030103 quotient, which is the log. of 1.07177, the root required. EXAMPLE III. Required the cube root of .00048 log. of .00048 4.681241 of it. 2.893747 = log. of .078297, the root. TABLE OF LOGARITHMIC, SINES, TANGENTS, &c. 56. This is Table II. It contains the logarithm of the sine, tangent, cosine and cotangent, corresponding to every degree and minute in the quadrant.* These logarithms are those of the trigonometrical lines in a circle, the radius of which is 10000000000, or the tenth power of 10, the common logarithm of which is 10. As the sine is never greater than radius, its logarithm will always be less than 10, except for the arc 90°, the sine of which is equal to 10. PROBLEM. 57. To find from the table the logarithm of the sine, tangent, cosine or cotangent of the number expressing any arc. CASE 1. If the given number is composed of degrees and * Without this table we should have been obliged to employ the two other tables which have been already described as follows. First we must have found the natural sine, tangent, &c. of the given arc or angle, in Table III., then with this have entered Table I. and found its logarithm. |