-:

then say, as 10 is to the difference, so is the fifth figure of the given number, to a fourth, which added to the less of the two logs., will give the log. sought.

As 10:17:: 6: 10.2, the 4th number.

The difference of the logs. is found in the right hand column of some tables, which saves the trouble of subtraction.

The above stating is founded on the supposition, that the differences of logs. are as the differences of the corresponding numbers. Thus,

Then 100:8:: 47: 3.76, or 4 nearly.

Then .73743+4=.73747; hence the log. of 546347 is 5.73747.

The calculator will soon perceive that it is unnecessary to state the numbers as above directed, it being sufficient to multiply the difference between the log. of the first four figures and the next log. greater, by the two remaining figures, and cutting off as many figures from the right hand of the product as are equal to the number of figures multiplied by.

To find the log. of a fraction; suppose 117.

Subtract the log. of the denominator from the log. of the numerator, and the remainder will be the log. of the fraction.

117

log. 2.06819

147

log. 2.16732

-1.90087 log. 117.

The fraction may be reduced to a decimal, and the log. found, as if whole numbers, except the index. If the significant figure be in the place of tenths, the index will be -1; if in the place of hundredths, it will be-2; if in the place of thousandths, it will be -3; and so on.

Thus, the log. of the log. of

.3754 is-1.57449 .03754 is-2.57449

the log. of .003754 is-3.57449, &c.

The log. of a mixed number is found as that of a whole number, except the index, which must be always 1 less than the number of places in the integral part.

Thus, the log. of 59684 is 4.77586
And the log. of 59.684 is 1.77586
Also the log. of .59684 is-1.77586.

To find the number answering to any logarithm to four places of figures.

This process is only the converse of finding the log. answering to a given number. Therefore, look for the given log. among the columns containing the logs., the number in the left hand column will be the three first figures, and the figure at the top will be the fourth figure; the integral part is to be regulated by the index of the given log., as in the last case.

Thus, the number answering to the log. 2.32716 is 312.4

The number answering to the log. 4.35005 is 22390. If the given log. cannot be found exactly in the tables, take the difference of the logs. next greater and next less, and also the difference between the given log. and the next less; then say, as the first of these differences is to the second, so is 10 to the fifth figure of the required number.

If the number be required to six places of figures, make 100 the third term of the proportion, and the figures thus formed, when annexed to the number answering to the next less log., will give the number sought.

Let it be required to find the number answering to the log. 2.26589.

Next less 2.26576, log. of 184.4

13, second difference.

As 24: 13 :: 10 : 5, the fifth figure. Hence the number required to five places of figures is 184.45.

When the number answering to the above log. is required to six places of figures, say, as 24:13::100:53. Hence the number required to six places is 184.453. Note.-When the logs. next less and next greater, fall in the latter part of the tables, where the differences are very small, the number answering to the given log. cannot be always depended on to more than five places of figures.

In treating of logs. having negative indices, we might have mentioned that the negative sign is put over the index, in order to distinguish it from the decimal part found in the tables, which is always positive: so -2 +69897, which is the log. of .05, is written -2.69897. It may be necessary to observe further, that on some occasions it is convenient to reduce the whole expression to a negative form, which is done by making the characteristic less by 1, and taking the arithmetical

complement of the decimal part of the log., that is, beginning at the left hand, subtract each figure from 9, except the last significant figure, which is subtracted from 10, then shall the remainders express the log. wholly negative. Thus the log. of .05, which is -2.69897, is expressed -1.30103, which is all negative.

Sometimes also it is thought more convenient to express such log. entirely as positive, by only joining to the tabular decimal the complement of the index to 10; and in this way the above log. is expressed by 8.69897, which is only increasing the index in the scale by 10.

MULTIPLICATION BY LOGARITHMS.

Multiplication is performed by adding the logs. of the factors together, and finding the natural number corresponding to the sum.

1. What is the product of 26 by 74?

26 log. 1.41497

74 log. 1.86923

1924 log. 3.28420

2. What is the product of .0054 by .95?

If we choose to avoid negative indices, each of the

factors, when decimals, may be multiplied by 10,