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To find, from the table, the logarithm of any number.

7. If the number is less than 100, look on the first page of the table, in the column of numbers under N, until the number is found: the number opposite is the logarithm sought: Thus,

log 9 = 0.954243.

When the number is greater than 100 and less than 10000.

8. Find in the column of numbers, the first three figures of the given number. Then pass across the page along a horizontal line until you come into the column under the fourth figure of the given number: at this place, there are four figures of the required logarithm, to which, two figurestaken from the column marked 0, are to be prefixed.

If the four figures already found stand opposite a row of six figures in the column marked 0, the two left hand figures of the six, are the two to be prefixed; but if they stand opposite a row of only four figures, you ascend the column till you find a row. of six figures; the two left hand figures of this row are the two to be prefixed. If you prefix to the decimal part thus found, the character. istic, you will have the logarithm sought: Thus,

log 8979 = 3.953228

log .08979= 2.953228 If, however, in passing back from the four figures found, to the 0 column, any dots be met with, the two figures to be prefixed must be taken from the horizontal line di. rectly below: Thus,

log 3098 = 3.491081

log 30.98=1.491081 If the logarithm falls at a place where the dots occur, O must be written for each dot, and the two figures to be prefixed are, as before, taken from the line below: Thus,

log 2188 = 3.340047
log .2188 = 1.340047

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When the number exceeds 10,000. 9. The characteristic is determined by the rules already given. To find the decimal part of the logarithm: place à decimal point after the fourth figure from the left hand, converting the given number into a whole number and decimal. Find the logarithm of the entire part by the rule just given, then take from the right hand column of the page, under D, the number on the same horizontal line with the logarithm, and multiply it by the decimal part; add the product thus obtained to the logarithm already found, and the sum will be the logarithm sought.

If, in multiplying the number taken from the column D, the decimal part of the product exceeds .5, let 1 be added to the entire part; if it is less than .5, the decimal part of the product is neglected.

EXAMPLE.
1. To find the logarithm of the number 672887.

The characteristic is 5.; placing a decimal point after the fourth figure from the left, we have 6728.87. The decimal part of the log 6728 is .827886, and the corresponding number in the column D is 65; then 65%.87= 56.55, and since the decimal part exceeds .5, we have 57 to be added to .827886, which gives .827943. · Hence, log 672887=5.827943

Similarly, log .0672887 = 2.827943

The last rule has been deduced under the supposition that the difference of the numbers is proportional to the difference of their logarithms, which is sufficiently exact within the narrow limits considered.

In the above example, 65 is the difference between the logarithm of 672900 and the logarithm of 672800, that is, it is the difference between the logarithms of two numbers which differ by 100. We have then the proportion

100 : 87 :: 65 : 56.55, hence, 56.55 is the number to be added to the logarithm before found.

To find from the table the number corresponding to a given

logarithm. 10. Search in the columns of logarithms for the deciinal part of the given logarithm: if it cannot be found in the table, take out the number corresponding to the next less logarithm and set it aside. Subtract this less logarithm from the given logarithm, and annex to the remainder as many zeros as may be necessary, and divide this result by the corresponding number taken from the column marked D, continuing the division as long as desirable : annex the quotient to the number set aside. Point off, from the left hand, as many integer figures as there are units in the characteristic of the given logarithm increased by 1; the result is the required number.

If the characteristic is negative, the number will be entirely decimal, and the number of zeros to be placed at the left of the number found from the table, will be equal to the number of units in the characteristic diminished by 1.

This rule, like its converse, is founded on the supposition that the difference of the logarithms is proportional to the difference of their numbers within narrow limits.

EXAMPLE.

1. Find the number corresponding to the logarithm 3.233568.

The decimal part of the given logarithm is .233568

The next less logarithm of the table is .233504, and its corresponding number 1712. Their difference is

. 64 Tabular difference

253)6400000(25 Hence, the number sought . 1712.25.

The number corresponding to the logarithm 3.233568 is .00171225.

2. What is the number corresponding to the logarithin 2.785407?

Ans. .06101084. 3. What is the number corresponding to the logarithm 7.846741:

Ans. .702653.

MULTIPLICATION BY LOGARITHMS. 11. When it is required to multiply numbers by means of their logarithms, we first find from the table the logarithms of the numbers to be multiplied; we next add these logarithms together, and their sum is the logarithm of the product of the numbers (Art. 3).

The term sum is to be understood in its algebraic sense; therefore, if any of the logarithms have negative characteristics, the difference between their sum and that of the positive characteristics, is to be taken ; the sign of the remainder is that of the greater sum.

EXAMPLES

1. Multiply 23.14 by 5.062.

log 23.14=1.364363

log 5.062 = 0.704322 Product, 117.1347 ... 2.068685

2. Multiply 3.902, 597.16, and 0.0314728 together.

log 3.902 = 0.591287
log 597.16=2.776091

log 0.0314728=2.497936
Product, 73.3354 .. .. 1.865314

Here, the 2 cancels the + 2, and the 1 carried from the decimal part is set down.

3. Multiply 3.586, 2.1046, 0.8372, and 0.0294 together.

log 3.586 = 0.554610
log 2.1046 = 0.323170
log 0.8372 = 1.922829

log 0.0294=2.468347
Product, 0.1857615 . . 7.268956

In this example the 2, carried from the decimal part, cancels 2, and there remains 1 to be set down.

DIVISION OF NUMBERS BY LOGARITHMS. 12. When it is required to divide numbers by means of their logarithms, we have only to recollect, that the subtraction of logarithms corresponds to the division of their numbers (Art. 4). Hence, if we find the logarithm of the dividend, and from it subtract the logarithm of the divisor, the remainder will be the logarithm of the quotient.

This additional caution may be added. The difference of the logarithms, as here used, means the algebraic difference; so that, if the logarithm of the divisor have a negative characteristic, its sign must be changed to positive, after diminishing it by the unit, if any, carried in the subtraction from the decimal part of the logarithm. Or, if the characteristic of the logarithm of the dividend is negative, it must be treated as a negative number.

EXAMPLES.

1. To divide 24163 by 4567.

log 24163 = 4.383151

log 4567=3.659631 Quotient, 5.29078 .. 0.723520

2. To divide 0.06314 by .007241.

log 0.06314 = 2.800305

log 0.007241 = 3.859799 Quotient, 8.7198 .0.940506

Here, 1 carried from the decimal part to the 3, changes it to ], which being taken from 2, leaves 0 for the characteristic.

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