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Now if we add T+1 to the

in

afterwards reduced to 2. the present instance, it will become + 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.

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In all cases of extracting roots of fractions, if the negative charucteristic is not divisible by the number expressing the root, it must be made so in a similar manner.

If the characteristic were

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. .77298 Ans.

1.552668 = + 3.552068(4

T.888167

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 corresponding to expressed in decimals.

The logarithm of 2 being 0.30103, that of will be-0.30103.

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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 = 1.

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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 reduced to a decimal.

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The leg. of viz. 2.823909 is called the Arithmetic Comple ment 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, &e. 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.

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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.

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Instead of using the logarithm 7.568202 in its present form, add 10 to its characteristic and it becomes 9.568202.

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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

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We may observe that, in this way, when the first left hand figure is tenths, the characteristic, instead of being 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.

3. What is the 3d power of .378 ?

log. .378

9.577492

Multiply by

3

28.732476

log. .05401 nearly ans.

8.732476

Multiplying by 3 is the same as adding the number twice to itself. The characteristic becomes 28, but omitting two of the tens or 20, it becomes 8, which shows it to be the logarithm of a fraction whose first place is hundredths.

If it is required to find the 3d root of a fraction, it is easy to see, that having taken out the logarithm of the fraction, it will be necessary to add two tens to the characteristic, for it is then considered the third power of some other fraction, and in raising the fraction to that power, two tens would be subtracted.

In the last example the logarithm of the power is 8.732476, but in order to take its 3d root, it will be necessary to add the two tens which were omitted.

For the second root one ten must be previously added, and for the fourth root, three tens, &c.

4. What is the 3d root of .027 ?

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In dividing a whole number by a fraction, if 10 be added to the characteristic of the dividend, it cancels the 10 supposed to be added to the divisor. If both are fractions the ten in the one cancels it in the other; and if the dividend only is a fraction, the answer will of course be a less fraction. Consequently in division the results will require no alteration.

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