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The second form is inconvenient because it is negative, and also because in seeking the number corresponding to the logarithm, a fraction would frequently be found with decimals in the denominator. It would be much better that the whole fraction should be expressed in decimals. If the fraction is used in the decimal form, the logarithms may be used for them almost as easily as for whole numbers. Suppose it is required to find the logarithm of .5 or o.. .

log. 5- log. 10 =0.698970 — 1.=-1+.698970. Suppose it is required to find the logarithm of .05 or nós.

5 10.**8870_-24698970.

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log. 5 - log. 100 = 0.698970 -2=-2 +.698970. The logarithms of 10, 100, 1000, &c. always being whole numbers, we have the two parts distinct. The logarithm of .5 is the same as that of 5 except that it has the number 1 joined to it with the sign —, which is sufficient to distinguish it, and show it to be a fraction. The logarithm of .05 also is the same, except that -- 2 is joined to it. That is, the logarithm of the numerator is positive, and that of the denominator negative.

This negative number joined to the positive fractional part, serves as a characteristic, and is a continuation of the principle shown above; thus The log. 500 is 2.698970

1.698970

0.698970 66 66 .5 1.698970 66 66 .05 2.698970

The logarithm of a decimal is the same as that of a whole number expressed by the same figures, with the exception of the characteristic, which is negative for the fraction; being 1 when the first figure on the left is tenths, 2 when the first is hundredths, &c. It is convenient to write the sign over the characteristic thus, 1, 2, &c. It is not necessary to put the sign + before the fractional part, for this will always be understood to be positive.

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 1, when the first is hundredths, the characteristic is 2,

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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 I 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

T.361728 log. 74.75 Ans.

1.873611 Multiply 872 by .097. log. 872

2.940516 log. .097 . . 2.986772 log. 84.584 Ans. . . 1.927288 In adding the logarithms, there is 1 to cariy from the decimal to the units. This one is positive, because the decimal part is so. Multiply .857 by .0093 log..857

1.932981 log. .0093 .

3.968483

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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. .C-17

2.672098

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log. .0000915662 Ans..

5.961736 In order to be able to take the second from the first, I change the characteristic 2 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 log. .038 . .

2.579784

log. 19.3 121 Ans. .. 1.286503

What is the 3d power of .25? log. .25 . . T.397940

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 characteristïc 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 2. Now if we add T + 1 to the 2 in the present instance, it will become 33 + 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 i being positive may be joined to the fractional part.

log. .015625 7.193820 = 3 f. 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

7.552668 = 4 + 3.552668(4

log. .77298 Ans.

1.888167 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 1 corresponding to i expressed in decimals.

The logarithm of 2 being 0.30103, that of I 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 = J.

Reduce to a decimal.. . log. 5

0.69897 =-1+1.698970 log. 8

0.903090

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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 rnultiplication, that is, by adding the logarithms. This may be effected on the following principle.. To divide by 2 is the sarne as to multiply by 1 or .5. Dividing by 5 is the saine 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 1. First find the logarithm of is reduced to a decimal.

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The log. of í viz. 2.823909 is called the Arithmetic Comple ment of the log. of 15.

The arithmetic co:nplement 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 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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