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Finding a Logarithm.

consider them as integers, regarding the other places as decimals.

Care must be taken not to confound the decimal point thus introduced with the actual decimal point of the number, of which it is altogether independent.

Find, in the tables, the decimal logarithm corre‐ sponding to the integral part of the number thus pointed off; and also the difference between this logarithm and the one next above it, that is, the logarithm of the number which exceeds this integral part by unity; this difference is often given in the margin of the tables.

Multiply this difference by the decimal part of the number as last pointed off, and omit in the product as many places to the right as there are places in this decimal part of the number.

The product, thus reduced, being added to the decimal logarithm of the integral part of the number, is the decimal part of the required logarithm.

23. Corollary. This process for finding the decimal part of the logarithm of a number, which exceeds the limits of the tables, is founded on the following law, easily deduced from the inspection of the tables.

If several numbers are nearly equal, their dif ferences are proportional to the differences of their logarithms.

24. EXAMPLES.

1 Find the logarithm of 0·00325787.

Solution.

Number corresponding to Logarithm.

The characteristic is - 3, instead of which

may be written 10—37.

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For the decimal part, the number is to be written

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Ans. 0.27701.

2. Find the logarithm of 1·8924.

3. Find the logarithm of 757-823000. Ans. 887956.

4. Find the logarithm of 0.00041359.

Ans. 461657, or 6·61657

5. Find the logarithm of 0.12345.

Ans. 109149, or 9-09149.

6. Find the logarithm of 99998.

Ans. 4.99999.

25. Problem. To find the number corresponding to a given logarithm.

Solution. First. In finding the figures of the required number, the characteristic is to be neglected.

Number corresponding to Logarithm.

When the decimal part of the given logarithm is exactly contained in the tables, its corresponding number can be immediately found by inspection.

But when the given logarithm is not exactly contained in the tables, the number, corresponding to the logarithms of the table which is next below it, gives the four first places on the left of the required number.

One or two more places are found by annexing one or two cyphers to the difference between the given logarithm and the logarithm of the tables next below it, and dividing by the difference between the logarithm of the tables next below and that next above the given logarithm.

When tables are used in which the logarithms are given to five places, the accuracy of the corresponding numbers is never to be relied upon to more than 6 places, and rarely to more than 5 places; so that in finding the last quotient, one place is usually sufficient.

Secondly. The position of the decimal point of the required number depends altogether upon the characteristic of the given logarithm, and is easily ascertained by the rule of art. 20; cyphers being prefixed or annexed when required.

26. EXAMPLES.

1. Find the number, whose logarithm is 8.19325. Solution. We have for the logarithm of the tables nex below the given logarithm

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Multiplication of Logarithms.

Hence

the diff. between given log. and log. 1560 = 13,

also

and the quotient

log. 1561 log. 1560 = 28,

138046

gives the two additional places; so that the six places of the

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2. Find the number, whose logarithm is 2-13511.

Ans. 136-493.

3. Find the number, whose logarithm is 1.76888.

Ans. 58-7328.

4. Find the number, whose logarithm is 0-11111.

Ans. 1.29153.

5. Find the number, whose logarithm is 2-98357.

Ans. 0.0962875.

6. Find the number, whose logarithm, when written 10 more than it should be, is 9.35846.

Ans. 0 22828.

27. Problem. To find the product of two or more factors by means of logarithms.

Solution. Find the sum of the logarithms of the factors, and the number, of which this sum is the logarithm, is, by art. 10, the required product.

When the logarithm of any of the factors is written, as in art. 22, 10 more than its true value, as many times 10 should be subtracted from the result as there are such logarithms.

Involution by Logarithms.

28. EXAMPLES.

1. Find the continued product of 78·052, 0·6178, 341000, 100-008, and 0.0009.

Solution. We find, from the tables,

log. 78-052 1.89238

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In the sum of the preceding logarithms 20 was neglected, because two of the logarithms were written 10 more than they should be.

2. Find the continued product of 0·0001, 7,9004, 0·56, 0 032569, and 17899.1.

Ans. 0.257792.

3. Find the continued product of 3.1416, 0.559, and 64.01.

Ans. 112-41.

4. Find the continued product of 3-26, 0-0025, 025, and 0.003.

29. Problem.

Ans. 0-00000611257.

To find any power of a given

number by means of logarithms.

Solution. Multiply the logarithm of the given number by the exponent of the required power, and

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