by 3, and the 1 being positive may be joined to the fractional part. log. .015625 2.1938203 +1.193820(3 log. .25 Ans. 1.397940 In all cases of extracting roots of fractions, if the negative characteristic is not divisible by the number expressing the root, it must be made so in a similar manner. If the characteristic were 3 and it were required to find the fifth root, we must add 2+2 and it will become 5 + 2. What is the 4th root of .357 ? log. .357 log. .77294 Ans. 7.5526684 +3.552668(4 1.888147 Any common fraction may be changed to a decimal by its logarithms, so that when the logarithm of a common fraction is required, it is not necessary to change the fraction to a decimal previous to taking it. It is required to find the logarithm of ing to expressed in decimals. correspond The logarithm of 2 being 0.30103, that of will be -0.30103. Now 0.30103- 1+1 .30103 =-1+(1 The decimal part .69897 is the log. of 5, and — 1 is the log. of 10 as a denominator. Therefore 1.69897 is the log. of .5 }· When there are several multiplications and divisions to be performed together, it is rather more convenient to perform the whole by multiplication, that is, by adding the logarithms. This may be effected on the following principle. To divide by 2 is the same as to multiply by or .5. Dividing by 5 is the same as multiplying by or .2, &c. Suppose then it is required to divide 435 by 15. Instead of dividing by 15 let us propose to multiply by First find the logarithm of a decimal. reduced to a 1.462398 log. 29 quotient of 435 by 15 The log. of viz. 2.823909 is called the Arithmetic Complement of the log of 15. The arithmetic complement is found by subtracting the logarithm of the number from the logarithm of 1, which is zero, but which may always be represented by i + 1, 2+2 &c. It must always be represented by such a number that the logarithm of the number may be subtracted from the positive part. That is, it must always be equal to the characteristic of the logarithm to be subtracted, plus 1; for 1 must always be borrowed from it, from which to subtract the fractional part. It is required to find the value of x in the following equation. I multiply by 3 to find the 3d power, and divide by 5 to obtain the 5th root. LI. There is an expedient generally adopted to avoid the negative characteristics in the logarithms of decimals. I shall explain it and leave the learner to use the method he likes the best. Instead of using the logarithm 1.568202 in its present form, add 10 to its characteristic and it becomes In this case 10 was added to one of the numbers and afterwards subtracted from the result; of course the answer must be the same. LI We may observe that, in this way, when the first left hand figure is tenths, the characteristic, instead of being ī is 9, and when the first figure is hundredths, the characteristic is 8 &c. That is, the place of the first figure of the number reckoned from the decimal point corresponds to what the characteristic falls short of 10. Whenever in adding, the characteristic exceeds 10, the ten or tens may be omitted and the unit figure only retained. In the first example, one number only was a fraction, viz. .37. In adding, the characteristic became 11, and omitting the 10 it became 1, which shows that the product is a number exceeding 10. In the second example both numbers were fractions, of course each characteristic was 10 too large. In adding, the characteristic became 18. Now instead of subtracting both tens or 20, it is sufficient to subtract one of them, and the characteristic 8, which is 2 less than 10, shows as well as 2 would do, that the product is a fraction, and that its first figure must be in the second place of fractions or hundredth's place. If three fractions were to be multiplied together, there would be three tens too much used, and the characteristic would be between 20 and 30; but rejecting two of the tens, or 20, the remaining figure would show the product to be a fraction, and show the place of its first figure. |