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In subtracting, the negative quantity is to be added, as in algebraic quantities.
In order to be able to take the second from the first, I change the characteristic 3 into T3 + 1 which has the same value. This enables me to take 9 from 18, that is, it furnishes a ten to borrow for the last subtraction of the positive part. In subtracting, the characteristic 2 of the second logarithm becomes
negative and of course must be added to the other negative. Divide .735 by .038.
The logarithm of this number is 3.193820. This characteristic 3 cannot be divided by 3, neither can it be joined with the first decimal figure in the logarithm, because of the different sign. But if we observe the operation above in finding the power, we shall see, that in multiplying the decimal part there was 1 to carry, which was positive, and after the multiplication was completed, the characteristic stood thus, 3 + 1 which was
afterwards reduced to 3. Now if we add T + 1 to the T in the present instance, it will become 3 + 1, and at the same time its value will not be altered. The negative part of the characteristic will then be divisible by 3, and the 1 being positive may be joined to the fractional part.
In all cases of extracting roots of fractions, if the negative chaacteristic 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.
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 3 corresponding to # expressed in decimals,
The logarithm of 2 being 0.30103, that of 4 will be —0.30103.
The decimal part .69897 is the log. of 5, and — 1 is the log, of 10 as a denominator. Therefore T.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 P. First find the 'ogarithm of to reduced to a decimal.
log. 1 is 0 = − 2 + 2.000000 log. 15 subtract 1.176091 log. To in form of a decimal T2,823909 log. 435 add 2.638489 log. 29 = quotient of 435 by 15 1.462398
The log. of so, viz. 3.823909 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 T + 1, 3 + 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; %. 1 must always be borrowed from it, from which to subtract the fractional part.
It is required to find the value of w in the following equation. Q = (* X 28 × oft) 387 × 2.896
I multiply by 3 to find the 3d power, and divide by 5 to ob“ain 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.
1. Multiply 253 by 37.
log. .37 1.56S202 log. 253 2.403121 log. 93.61 nearly answer 1.97 1323
Instead of using the logarithm I.568202 in its present form, add 10 to its characteristic and it becomes 9.568.202.
log. .37 9.568202 log. 253 2.403121 II.97 1323
Subtract 10. log. 93.61 as above. 1.97 1323
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.
2. Multiply .023 by .976.
- .* Take out the logarithms of these numbers and add 10 to . . each characteristic, - *
log. .976 - 9.989.450 18.3511.78
Subtract 20 log.0224473 nearly ans. 2.3511.78
We may observe that, in this way, when the first left hand figure is tenths, the characteristic, instead of being T 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 ene of them, and the characteristic S, which is 2 less than 10, shows as well as 3 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.
log. .378 . o 6 e 9.577492