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DIVISION BY LOGARITHMS.

FROM the logarithm of the dividend, as found in the tables, subtract the logarithm of the divisor, and the natural number answering to the remainder, will be the quotient required.

Observing, if the subtraction cannot be made in the usual way, to add, as in the former rule, the 1 that is to be carried from the decimal part, when it occurs, to the index of the logarithm of the divisor, and then this result, with its sign changed, to the remaining index, for the index of the logarithm of the quotient.

EXAMPLES.

1. Divide 4768.2 by 36.954, by logarithms.

Nos.

4768.2

36.954

Logs. 3.6783545 1.5676615

Quot. 129.032 .

2. Divide 21.754 by 2.4678, by logarithms.

Nos. 21.754

2.4678

Quot. 8.1518 .

3. Divide 4.6257 by .17608, by logarithms.

Nos. 4.6257

.17608

2.1106930

Logs. 1.3375391 0.3923100

.27684
5.1576

0.9452291

4. Divide .27684 by 5.1576, by logarithms.

Nos.

Logs. 0.6651725

1.2457100

Quot. 26.2741.

1.4194625

Here — 1, in the lower index, is changed into +1, which is then taken for the index of the result.

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Logs.

1.4422288
0.7124477

Quot. .0536761

1.7297811

Here the 1 that is to be carried from the decimals, is taken

as

1, and then added to 1, in the upper index, which gives 2 for the index of the result.

5. Divide 6.9875 by .075789, by logarithms.

Nos. 6.9875.

.075789

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Quot. 92.1967

Here the 1, that is to be carried from the decimals, is added 1, and this put down, with its sign

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to 2, which makes changed, is +1.

6. Divide .19876 by .0012345, by logarithms.

Nos.

Logs. 1.2983290

.19786 .

.001.2345

3.0914911

Logs. 0.8443218

2.8796062

1.9647156

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Quot. 161.0051 . 2.2068379

Here 3, in the lower index, is changed into +3, and this added to 1, the other index, gives + 3 — 1 or 2. 7. Divide 125 by 1728, by logarithms. Ans. 0723379. 8. Divide 1728.95 by 1.10678, by logarithms.

Ans. 1562.144.

9. Divide 10.23674 by 4.96523, by logarithms.

10. Divide 19956.7 by .048235, by logarithms

11. Divide .067859 by 1234.59, by logarithms. Ans. .0000549648.

Ans. 2.061685.

Ans. .413739.

THE RULE OF THREE, OR PROPORTION,
BY LOGARITHMS.

FOR any single proportion, add the logarithms of the second and third terms together, and subtract the logarithm of the first from their sum, according to the foregoing rules; then the natural number answering to the result will be the fourth term required.

Bu if the proportion be compound, add together the logarithms of all the terms that are to be multiplied, and from the result take the sum of the logarithms of the other terms, and the remainder will be the logarithm of the term sought.

Or, the same may be performed most conveniently thus :--Find the complement of the logarithm of the first term of the proportion, or what it wants of 10, by beginning at the lefthand, and taking each of its figures from 9, except the last significant figure on the right, which must be taken from 10; then add this result and the logarithms of the other two terms together, and the sum, abating 10 in the index, will be the logarithm of the fourth term, as before.

And if two or more logarithms are to be subtracted, as in the latter part of the above rule, add their complements and the logarithms of the terms to be multiplied together, and the result, abating as many 10s in the index as there are logarithms to be subtracted, will be the logarithm of the term required; observing when the index of the logarithm, whose complement is to be taken, is negative, to add it, as if it were affirmative, to 9; and then take the rest of the figures from 9, as before.

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1. Find a fourth proportion to 37.125, 14.768, and 135,279, by logarithms.

Log. of 37.125
Complement
Log. of 14.768
Log. of 135.279

Complement
Log. of .7186

Log. of .34721

EXAMPLES.

Ans. 53.81099

1.7308856

2. Find a fourth proportional to .05764, .7186, and .34721, by logarithms.

Log. of .05764

2.7607240

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11.2392760

1.8564872

1.5405992

0.6363554

3. Find a third proportional to 12.796, and 3.24718, by logarithms.

Log. of 12.796

1.1070742

Complement
Log, of 3.24718

Ans. 4.328681

1.5696665

8.4303335

1.1693217

2.1312304

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8.8929258
0.5115064

Log. of 3.24718

Ans. .8240216

1.9159386

4. Find the interest of 2797. 5s. for 274 days, at 41 per cent. per annum, by logarithms.

Comp. log. of 100
Comp. log. of 365
Log. of 279.25
Log. of 274
Log. of 4.5

Ans. 9.433296 '.

0.5115064

8.0000000

7.4377071

2.4459932

2.4377506

0.6532125

0.9746634

5. Find a fourth proportional to 12.678, 14.065, and 100.979, by logarithms.

Ans. 112.0263.

6. Find a fourth proportional to 1.9864, .4678, and 50.4567, by logarithms. Ans. 11.88262.

7. Find a fourth proportional to .09658, 24958, and .005967, by logarithms. Ans. .02317234.

8. Find a third proportional to .498621, and 2.9587, and a third proportional to 12.796, and 3.24718, by logarithms.

Ans. 17.55623, and .8240216.

INVOLUTION, OR THE RAISING OF POWERS BY LOGARITHMS.

EXAMPLES.

TAKE out the logarithm of the given number from the tables, and multiply it by the index of the proposed power; then the natural number answering to the result, will be the power required.

Observing, if the index of the logarithm be negative, that this part of the product will be negative; but as what is to be carried from the decimal part will be affirmative, the index of the result must be taken accordingly.

1. Find the square of 2.7568, by logarithms. Log. of 2.7568

0.4402477

Square 7.599946

2. Find the cube of 7.0851, by logarithms.

0.8804954

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Cube 355.6475 .

2.5510197

3 Find the fifth power of .87451, by logarithms. Log. of .87451

1.9417648

3

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Fifth power .5114695. 1.7088240

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Where 5 times the negative index 1, being +4 to carry, the index of the power is 1. 4. Find the 365th power of 1.0045, by logarithms.

Log. 1.0045*

0.0019499
365

5

58497

97495 116994

07117135

5, and

Power 5.148888

5. Required the square of 6.05987, by logarithms. Ans. 36.72603. 6. Required the cube of .176546, by logarithms.

Ans. 005502674.

7. Required the 4th power of .076543, by logarithms. Ans. 0000343259. by logarithms.

8. Required the 5th power of 2.97643,

Ans. 233.6031. by logarithms.

9. Required the 6th power of 21.0576,

Ans. 87187340.

10. Required the 7th power of 1.09684, by logarithms. Ans. 1.909864.

* The answer, 5.148888, though found strictly according to the general rule, is not correct in the last four figures, 8888; nor can the answers to such questions, relating to very high powers, be generally found true to 6 places of figures by the tables of Log. commonly used; if any power above the hundred thousandth were required, not one figure of the answer here given could be depended on. The Log. of 1.0045 is 00194994108 true to eleven places, which, multiplied by 365, gives .7117285 true to 7 places, and the corresponding number true to 7 places is 5.149067. See Doctor Adrain's edition of Hut. Math. Vol. 1. p. 169.

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