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Here the 2, to be carried, cancels the 2, and there re. mains the I to be set down.

DIVISION BY LOGARITHMS.

RULE.

From the logarithm of the dividend subtract the logarithm of the divisor, and the number answering to the remainder will be the quotient required.

NOTE. If i be to be carried to the index of the subtrahend, apply it according to the sign of the index; then change the sign of the index to -, if it be t, or to ti if it be ; and proceed according to the second note una der the last rule.

EXAMPLES.

1. To divide 24163 by 45677
Num.

Log
Dividend 24163

43831509
Divisor
4567

36596310

Quotient 5*290782

097235199

ܕ!,x ܀ ܕ

2. To divide 37*149* by 523*76.
Num.

Log.
Dividend 37:149

1'5699471
Divisor : 523.76 2'7191323
Quotient *07092752 ---28508148

:

3. Divide

3. Divide .06314 by ‘007245.
Num.

Log.
Dividend '06314 -2.8003046
Divisor '007241 3.8597985

Quotient 8.719792 o'9405061 Here 1, carried from the decimals to the 3, makes it become-2, which, taken from the other 2, leaves o remaining. 4. To divide •7438 by 12'9476. Num.

Log.
Dividend "7438

1.8714562
Divisor 12'9476

1'1121893

Quotient '05744694

-2.7592669

Here the 1, taken from the 1, makes it become to be set down.

INYOLUTION BY LOGARITHMS.

RULE.

Multiply the logarithm of the given number by the index of the power, and the number answering to the product will be the power required.

Note. A negative index, multiplied by an' affirmative number, gives a negative product ; and as the number carried from the decimal part is affirmative, their difference with the sign of the greater is, in that case, the index of the product

EXAMPLES EXAMPLES

1. To square the number 2'5791.
Num.

Log.
Root 2-5791

0-4114682
The Index

2

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2. To find the cube of 3*07146.
Num.

Log.
Root 3*07146

0*4873449
The Index

3

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Log

To raise *09163 to the 4th power.

Num.
Root .09163 -2.9620377
The Index

4

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Here 4 times the negative index being -8, and 3 to be carried, the difference - 5 is the index of the product.

4
To raise 1'0045 to the 365th root.
Num.

Log.
Root I'0045

0:0019499
The Index

365

97495 116994 58497

Power 5*148888

*7117135

EVOLUTION 0'0000524

EVOLUTION BY LOGARITHMS.

RULE.

Divide the logarithm of the given number by the index of the power, and the number answering to the quotient will be the root required.

Note. When the index of the logarithm is negative, and cannot be divided by the divisor without a remainder, increase the index by a number, that will render it exactly divisible, and carry the units borrowed, as so many tens, to the first decimal place; and divide the rest as usuala

EXAMPLES

1. To find the square root of 365.
Num.

Log
Power 365

2)2 5622929.
Root 19'10498

1'2811465

2. To find the 3d root of 12345.
Num.

Log.
Power 12345

3/4'0914911
Root 23*11162

1*3638304

3. To find the roth root of 2,
Num.

Log.
Power
2

10)0*3010300
Root 1'071773

o'0301030

4. To find the 365th root of 1'045.
Num.

Log.
Power 1'045

365)0:0191163
Root 1000121

5. To find the second root of '093. Num.

Log

2)-2.9684829 Root '304959

-I'4842415

Power '093

Here the divisor 2 is contained exactly once in the negative index -2, and therefore the index of the quotient is

-I.

6. To find the third root of '00048.
Num.

Log.
Power '00048 3-46812412
Root '07829735

-2.8937471

Here the divisor 3 not being exactly contained in 4, 4 is augmented by 2, to make up 6, in which the divisor is contained just 2 times; then the 2, thus borrowed, being carried to the decimal figure 6, makes 26, which, divided by 3, gives 8, &c.

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