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rithm of the numerator, the remainder is the loga. rithm of the fraction.

EXAM. What is the logarithm of 178 ?

From the log. of 179 (add 10

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5. To find the logarithm of a decimal fraction, pro. ceed as if it were an integer, and prefix the proper index.

6. To find the number anfwering to any given logarithm.

Look for the fractional part of the given logarithm in the different columns until you find it, either exactly, or the next lefs. And, in a line with the logarithm found, in the column marked No. you have three figures of the number fought, and on the top of the page you have one figure more, which write down, and point it according to the index of the given logarithm.

If the logarithm is not found exactly in the tables, and more than four figures are required, subtract the logarithm found in the table from the given one, and divide the remainder by the difference between the logarithm found and the next greater; the quotient joined to the four figures already found, gives the number anfwering to the logarithm.

EXAMPLES.

To find the number anfwering to the logarithm 2.7342957. Seek in the fecond column for the three figures next the index (-734), and, having found them, look among the columns on the right hand for the laft four figures, and in that under 3, you find 2396; therefore the number is 5423; but, because the index of the logarithm is 2, the number answering hath only three integer figures, and the last is a decimal, viz. 542.3.

If more figures are desired, divide the difference between 2957 and 2396, viz. 561, by 801, the difference of the logarithms (annexing o to 561), the quotient 7, joined to the former, makes the answer 542.37. Required the numbers anfwering to thefe loga. rithms.

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7.To perform multiplication by logarithms, add the logarithm of the multiplier to the logarithm of the multiplicand; the fum is the logarithm of the product,

EXAM

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4.2507371

Product 17813

2. Multiply 79.8 by 2.79.

8. To perform divifion by logarithms.

Subtract the logarithm of the divifor from the logarithm of the dividend, the remainder is the loga.

rithm of the quotient.

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9. To find a fourth proportional to three given numbers, or to work the rule of three by logarithms:

Add the logarithms of the second and third terms, and fubtract the logarithm of the first term from the fum; the remainder is the logarithm of the fourth term, or answer.

EXAM. If 497 yards coft L. 287; what will 389 Anf. L. 224: 12:8.

yards coit?

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N. B. Inftead of fubtracting the logarithm of the firft term, fome chuse to add its arithmetical comple

ment.

10. To find the arithmetical complement of a logarithm; begin at the left hand, and write down what each figure wants of 9, and what the laft fignificant figure wants of 10; fo the arith. comp. of 2.6963564 is 7.3036436.

11. To raise a number to any power, by logarithms. Multiply the logarithm of the given number by the index of the power; the product is the logarithm of the power required.

EXAM. Required the third power or cube of 81 ?

81 its log. is 1.9084850

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Anf. 531441 5.7254550.

2. Required the fourth power of 1.05? 1.05 its logarithm is 0.0211893 Multiply by

Anf. 1.215506

4

0.0847572

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Here the tens are thrown away from the index 52, and the 2 remaining fhews, that the firft fignificant

figure must be in the eighth place from unity, and therefore seven cyphers must be prefixed.

12. To extract roots by logarithms.

Divide the logarithm of the given number by the index of the power; the quotient is the logarithm of the root; that is, divide by 2 for the fquare root, by 3 for the cube root, and by 4 for the biquadrate, &c. EXAM. It is required to extract the fquare root, cube root, and biquadrate root of 19?

Logarithm of 19 is 1.2787536

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.3196884 gives 2.0877 biq. root.

2. It is required to extract the cube root of .78547 Log. of .7854 is 9.8950909; prefix 2 to the index.

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In extracting the roots of decimal fractions by lo. garithms; prefix to the index of the log. a number lefs by one than the index of the root, and then divide as before.

Of Logarithm Sines and Tangents.

The table of fines and tangents has the go of the quadrant difpofed thus. At the head of the two firft

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