Which logarithm is also correct to the nearest unit in the last figure.

And in the same way we may proceed to find the logarithm of any prime number.

Also, because the sum of the logarithms of any two numbers gives the logarithm of their product, and the difference of the logarithms the logarithm of their quotient, &c.; we may readily compute, from the above two logarithms, and the logarithm of 10, which is 1, a great number of other logarithms, as in the following examples :

3. Because 2X2=4, therefore log. 2 .301029995

And thus, by computing, according to the general formula, the logarithms of the next succeeding prime numbers 7, 11, 13, 17, 19, 23, &c. we can find, by means of the simple rules, before laid down for multiplication, division and the raising of powers, as many other logarithms as we please, or may speedily examine any logarithm in the table.

MULTIPLICATION BY LOGARITHMS.

Take out the logarithms of the factors from the table, and add them together; then the natural number answering to the sum will be the product required.

Observing, in the addition, that what is to be carried

from the decimal part of the logarithms is always affirmative, and must, therefore, be added to the indices, or integral parts, after the manner of positive and negative quantities in algebra.

Which method will be found much more convenient, to those who possess a slight knowledge of this science, than that of using the arithmetical complements.

Here, the+1, that is to be carried from the decimals, cancels the the upper line to be set down.

1, and consequently there remains 1 in

Here the +1 that is to be carried from the decimals, 1, in the upper line, as before, and there 2 to be set down.

destroys the remains the

5. Multiply 3.768, 2.053, and .007693, together.

Here the +1, that is to be carried from the decimals, when added to -3, makes 2, to be set down.

6. Multiply 3.586, 2.1046, .8372, and .0294, together.

Here the +2, that is to be carried, cancels the -2,

there remains the 1 to be set down.

and

7. Multiply 23.14 by 5.062 by logarithms.

Ans. 117.1347

8. Multiply 4.0763 by 9.8432, by logarithms.

Ans. 40.12383

9. Multiply 498.256 by 41.2467, by logarithms.

Ans. 20551.41

10. Multiply 4.026747 by .012345, by logarithms.

Ans. .0497102

11. Multiply 3.12567, .02868, and .12379, together, by logarithms. Ans. 09109705 12. Multiply 2876.9, 10674, .098762, and .0031598, by logarithms. Ans. .0958299

DIVISION BY LOGARITHMS.

From the logarithm of the dividend, as found in the tables, subtract the logarithm of the divisor, and the natural number, answering to the remainder, will be the quotient required.

Observing, if the subtraction cannot be made in the usual way, to add, as in the former rule, the 1 that is to be carried from the decimal part, when it occurs, to the index of the logarithm of the divisor, and then this result, with its sign changed, to the remaining index, før the index of the logarithm of the quotient.