Multiplication of Logarithms.

Hence

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

also

and the quotient

log. 1561-log. 1560 = 28,

130° = 46

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

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 011111.

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

10+ log. 0.61789-79085

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, 0.25, 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

Evolution by Logarithms.

the number, of which this product is the logarithm, is, by art. 11, the required power.

When the logarithm of the given number is written 10 more than it should be, as many times 10 must be deducted from the product as there are units in the given exponent.

30. EXAMPLES.

1. Find the 4th power of 0.98573.

Solution. We have, by the tables,

10+ log. 0.985739.99375

multiply by

4

10+ log. 0.944069-97500

and the required power is

0.94406.

In the above product, 40 should have been neglected, but in order to avoid a negative characteristic, only 30 was neglected, leaving the exponent 10 too large.

2. Find the 3d power of 0.25.

Ans. 0.015625.

Ans. 3020-28.

3. Find the 7th power of 3.1416.

Ans. 0-00000987325.

4. Find the square of 0·0031422.

31. Problem.

To find any root of a given number by means of logarithms.

Solution. Divide the logarithm of the given number by the exponent of the required root, and the number, of which this quotient is the logarithm, is, by art. 12, the required root.

Evolution by Logarithms.

When the logarithm of the given number has a negative characteristic, instead of being increased by 10, it should be increased by as many times 10 as there are units in the exponent of the root, and the quotient will in this case exceed its true value by 10.

32. EXAMPLES.

1. Find the fifth root of 0.028145.

Solution. We have, by the tables,

50+ log. 0.028145 48-44940,

which, divided by 5, gives

33. The arithmetical complement of a logarithm is the remainder after subtracting it from 10.

34. Corollary. The arithmetical complement of the logarithm of a number is, by art. 14, and the preceding article, the logarithm of its reciprocal increased by 10.

35. Corollary. The most convenient method of finding the arithmetical complement of a logarithm is to subtract the first significant figure on the right from 10, and each figure to the left of this figure from 9.

Arithmetical Complement.

36. EXAMPLES.

1. Find the arithmetical complement of 9.62595.

Ans. 0.37405.

2. Find the arithmetical complement of the logarithm of 6. Ans. 9-22185.

3. Find the arithmetical complement of the logarithm of 0.07. Ans. 11.15490.

4. Find the reciprocal of 0-01115.
Solution. We have, by the tables,

log. 0-01115 (ar. co.) 11.95273
subtract

log. 89-686

10.

1.95273

37. Problem. To find the quotient of one number divided by another by means of logarithms.

Solution. Subtract the logarithm of the divisor from that of the dividend, and the number, of which the remainder is the logarithm, is, by art. 13, the required quotient.

Or, since, by art. 81, multiplying by the reciprocal of a number is the same as dividing by it, add the logarithm of the dividend to the arithmetical complement of the logarithm of this divisor, and the sum diminished by 10 is the logarithm of the quotient.