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In operating upon these numbers, the same rules must be ob

served as in other cases where numbers are found connected with the signs –H and —.

When the first figure of the fraction is tenths, the character1stic is T, when the first is hundredths, the characteristic is 2, &c.

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This is the same as the logarithm of 25, except that the characteristic T shows that its first figure on the left is 10ths, or one place to the right of units.

Multiply 325 by. 23.

log. 325 . . . . 2,511883
log. .23 . . . TT361728

log. 74.75 Ans. . . . 1.873611 Multiply 872 by .097.

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In adding the logarithms, there is 1 to carry from the decimal to the units. This one is positive, because the decimal part is so.

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log. 3000 Ans. e e o 3.477121

In subtracting, the negative quantity is to be added, as in algebraic quantities.

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Divide .076 by 830.
log. .076 TESS0814 = 3 + 1.880814

log. 830 - - 2.919078 log. .0000915662 Ans. - T5.961736

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.

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log. .735 . . - I866287
log. .038 . - 2.579784
log. 19.3421 Ans. - 1.286503
What is the 3d power of .25?
log. .25 - - T.397940
3
2. -
log. 0.015625 Ans. T + 1.193820 = 3.193820.

What is the third root of 0.015625 °

The logarithm of this number is 3.193820. This characteristic To 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 2. Now if we add T + 1 to the 3 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.

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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 5 and it were required to find the fifth root, we must add 2 + 2 and it will become 5 +2.

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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.

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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 = }.

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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 salne as multiplying by 4 or 2, &c.

Suppose then it is required to divide 435 by 15. Instead of dividing by 15 let us propose to multiply by T's. First find the logarithm of to reduced to a decimal.

log. 1 is 0 = — 2 + 2.000000 log. 15 subtract 1.176091 log. I'; in form of a decimal T2,823909 log. 435 add 2.638489 log. 29 = quotient of 435 by 15 1.462398

The log of P. viz. 3.823909 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 T + 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; #. I must always be borrowed from it, from which to subtract the fractional part.

It is required to find the value of a in the following equation.

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log. 35 1.544068 log. 28 1.447158 log. 56.78 1.754195 log. 387 2.587711 Arith. Com. T3412289 log. 2.896 0.461799 44 64 T53820.1 1.6959 II 3

5.087733(5 log. 10.4123 very nearly answer 1.017546

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. * 1. Multiply 253 by .37. log. .37 T.568202 log. 253 - - 2.403121 : log. 93.61 nearly answer 1.971323

Instead of using the logarithm I.568202 in its present form, add 10 to its characteristic and it becomes 9.568202.

log. .37 9.568202 log. 253 2.403121 11.971323

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.

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