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. 0·98573 = 9·99375.
multiply by

4

10+ log. 0·94406 = 997500

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.

4. Find the square of 0 0031422.

Ans. 0.00000987325.

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, i, 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 e quotient will in this case exceed its true value by 10.

1. Find the fifth root of 0.028145.

Solution. We have, by the tables,

50+ log. 0028145 48-44940,

which, divided by 5, gives

4. Find the square root of 238-149. Ans. 15 4317.

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

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.

Division by Logarithms.

When the logarithm of the dividend is written 10 more than its true value, 20 must be subtracted from the sum, instead of 10.

39. Corollary. The value of any fraction may be found by adding together the logarithms of all the factors of the numerator and the arithmetical complements of the logarithms of all the factors of the denominator, and subtracting from the sum as many times 10 as there are arithmetical complements plus as many times 10 as there are logarithms of the factors of the numerator, which are written greater than their true value by 10; the remainder is the logarithm of the fraction.

41. Corollary. The logarithm of the fourth term of a proportion is found by adding together the arithmetical complement of the logarithms of the first term and the logarithms of the second and third terms.