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In operating upon these numbers, the same rules must be observed as in other cases where numbers are found connected with the signs + and

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

The log. of.25 is log. 25 - log. 100
= 1.397940 -- 2=-2 +1.397940 = 1.397940.

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.
of ilog. 325982.511883 10,

T.361728

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bola log. 74.75 Ans.

1.873611 Multiply 872 by .097.

2.940516

2.986772 log..097 olaroni se s log. 84.584 Ans.

1.927288 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. Multiply .857 by .0093 log. .857

1.932981 vionw ss log. .009315 ori ob 3.968483 lo qoidq937. os log. .0079701 Ans.

3.901464 Divide 75 by .025. log. 75

1.875061 log. .025

2.397940

qog log, 3000 Ans.

26 2.477121

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In subtracting, the negative quantity is to be added, as in algebraic quantities. Divide 275 by .047. log. 275

2.439333 log. .047

2.672098

.

log. 5851.07 Ans.

3.767235 Divide .076 by 830.

log. .076 2.880814 = 3. + 1.880814
log. 830

2.919078

log. .0000915662 Ans.

5.961736 In order to be able to take the second from the first, I change the characteristic 7 into 3 + 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. log. .735

1.866287 2.579784

log. .038

log. 19.3422 Ans.

1.286503 What is the 3d power of .25? log. .25

T.397940

3

log. 0.015625 Ans. 3 + 1.193820 = 2.193820.

What is the third root of 0.015625 ?

The logarithm of this number is 2.193820. This characteristic 2 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 7. Now if we add 1 + 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.

log. .015625 2.193820 = 3 + 1.193820(3
log. .25 Ans.

1.397940 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 7 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

T.552668 = 4+ 3.552668(4

log. .77294 Ans.

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 1 corresponding to } expressed in decimals.

The logarithm of 2 being 0.30103, that of 4 will be —0.30103. Now

-0.30103 =-1+1-.30103 =-1+ (1 - .30103) =7.69897. 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 = 4.

Reduce to a decimal. log. 5

0.69897 -1 + 1.698970

0.903090

log. 8

log. 0.625 = Ans.

1.795880

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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 inultiplication, 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 is. First find the logarithm of 15 reduced to a decimal.

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The log. of i's 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 1 +1,"? + 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.

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log. 35 log. 28 log. 56.78 log. 337 log. 2.896

1.544068 1.447158 1.754195 3.412289 1.538201

2.587711 Arith. Com.
0.461799

1.695911

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 ob tain 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

1.568202 log. 253

2.403121

log. 93.61 nearly answer

1.971323 Instead of using the logarithm 1.568202 in its present form, add 10 to its characteristic and it becomes 9.568202.

log. .37 log. 253

9.568202 2.403121

11.971323 10.

Subtract

log. 93.61 as above.

1.971323 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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