EXAMPLES.

1. Let it be required to find the number corresponding to the logarithm 5.233568.

The next less mantissa in the table is 233504; the corresponding number is 1712, and the tabular difference is

.. The required mumber is 171225.296.

The number corresponding to the logarithm 2.233568 is .0171225.

2. What is the number corresponding to the logarithm 2.785407 ? Ans. .06101084.

3. What is the number corresponding to the logarithm 1.846741 ? Ans. .702653.

MULTIPLICATION BY MEANS OF LOGARITHMS.

15. From the principle proved in Art. 5, we deduce the following

RULE.

Find the logarithms of the factors, and take their sum; then find the number corresponding to the resulting logarithm, and it will be the product required.

2. Find the continued product of 3.902, 597.16, and 0.0314728.

Here, the 2 cancels the +2, and the 1 carried from the decimal part is set down.

3. Find the continued product of 3.586, 2.1046, 0.8372, and 0.0294. Ans. 0.1857615.

DIVISION BY MEANS OF LOGARITHMS.

16. From the principle proved in Art. 6, we have the following

RULE.

Find the logarithms of the dividend and divisor, and subtract the latter from the former; then find the number corresponding to the resulting logarithm, and it will be the quotient required.

The

Here, 1 taken from I, gives 2 for a result. subtraction, as in this case, is always to be performed in the algebraic sense.

3. Divide 37.149 by 523.76.

Ans. 0.0709274.

The operation of division, particularly when combined with that of multiplication, can often be simplified by using the principle of

17. The ARITHMETICAL COMPLEMENT of a logarithm is the result obtained by subtracting it from 10.

is the arithmetical complement of 1.869544.

Thus, 8.130456

The arithmetical

complement of a logarithm may be written out by commeneing at the left hand and subtracting each figure from 9,

until the last significant figure is reached, which must be The arithmetical complement is denoted by

taken from 10.

the symbol (a. c.).

Let α and b represent any two logarithms whatever, and α b their difference. Since we may add 10 to,

and subtract it from, ab, without altering its value, we have,

But, 106 is, by definition, the arithmetical complement of b: hence, Equation (10) shows that the difference between two logarithms is equal to the first, plus the arithmetical complement of the second, minus 10.

Hence, to divide one number by another by means of the arithmetical complement, we have the following

RULE.

Find the logarithm of the dividend, and the arithmetical complement of the logarithm of the divisor, add them toge ther, and diminish the sum by 10; the number correspond ing to the resulting logarithm will be the quotient required.

The operation of subtracting 10 is always performed mentally.

RAISING ΤΟ POWERS BY MEANS OF LOGARITHMS.

18. From the principle proved in Art. 7, we have the following

RULE.

Find the logarithm of the number, and multiply it by the exponent of the power; then find the number correspond ing to the resulting logarithm, and it will be the power required.