From the above examples, we see, that in a number composed of an entire and decimal part, we may change the place of the decimal point without changing the deci mal part of the logarithm; but the characteristic is dimin ished by 1 for every place that the decimal point is removed to the left.

In the logarithm of a decimal, the characteristic becomes negative, and is numerically 1 greater than the number of ciphers immediately after the decimal point. The negative sign extends only to the characteristic, and is written over it, as in the examples given above.

TABLE OF LOGARITHMS.

6. A table of logarithms, is a table in which are written the logarithms of all numbers between 1 and some given number. The logarithms of all numbers between 1 and 10,000 are given in the annexed table. Since rules have been given for determining the characteristics of logarithms by simple inspection, it has not been deemed necessary to write them in the table, the decimal part only being given. The characteristic, however, is given for all numbers less than 100.

The left hand column of each page of the table, is the column of numbers, and is designated by the letter N; the logarithms of these numbers are placed opposite them on the same horizontal line. The last column on each page, headed D, shows the difference between the logarithms of two consecutive numbers. This difference is found by subtracting the logarithm under the column headed 4, from the one in the column headed 5 in the same horizontal line, and is nearly a mean of the differ

To find, from the table, the logarithm of any number.

7. If the number is less than 100, look on the first page of the table, in the column of numbers under N, until the number is found: the number opposite is the logarithm ought: Thus,

log 90.954243.

When the number is greater than 100 and less than 10000.

8. Find in the column of numbers, the first three figures of the given number. Then pass across the page along a horizontal line until you come into the column under the fourth figure of the given number: at this place, there are four figures of the required logarithm, to which two figures taken from the column marked 0, are to be prefixed.

If the four figures already found stand opposite a row of six figures in the column marked 0, the two left hand figures of the six, are the two to be prefixed; but if they stand opposite a row of only four figures, you ascend the column till you find a row of six figures; the two left hand figures of this row are the two to be prefixed. If you prefix to the decimal part thus found, the characteristic, you will have the logarithm sought: Thus,

If, however, in passing back from the four figures found, to the 0 column, any dots be met with, the two figures to be prefixed must be taken from the horizontal line directly below: Thus,

If the logarithm falls at a place where the dots occur, 0 must be written for each dot, and the two figures to be prefixed are, as before, taken from the line below: Thus, log 21883.340047

When the number exceeds 10,000.

9. The characteristic is determined by the rules already given. To find the decimal part of the logarithm: place a decimal point after the fourth figure from the left hand, converting the given number into a whole number and decimal. Find the logarithm of the entire part by the rule just given, then take from the right hand column of the page, under D, the number on the same horizontal line with the logarithm, and multiply it by the decimal part; add the product thus obtained to the logarithm already found, and the sum will be the logarithm sought.

If, in multiplying the number taken from the column D, the decimal part of the product exceeds .5, let 1 be added to the entire part; if it is less than .5, the decimal part of the product is neglected.

EXAMPLE.

1. To find the logarithm of the number 672887. The characteristic is 5.; placing a decimal point after the fourth figure from the left, we have 6728.87. The decimal part of the log 6728 is .827886, and the corresponding number in the column D is 65; then 65X.87= 56.55, and since the decimal part exceeds .5, we have 57 to be added to .827886, which gives .827943.

Hence,
Similarly,

log 672887= 5.827943
log .0672887 = 2.827943

The last rule has been deduced under the supposition that the difference of the numbers is proportional to the difference of their logarithms, which is sufficiently exact within the narrow limits considered.

In the above example, 65 is the difference between the logarithm of 672900 and the logarithm of 672800, that is, it is the difference between the logarithms of two numbers which differ by 100.

..

We have then the proportion

100 : 87 :: 65 : 56.55,

Lence, 56.55 is the number to be added to the logarithm

To find from the table the number corresponding to a given

logarithm.

10. Search in the columns of logarithms for the decimal part of the given logarithm: if it cannot be found in the table, take out the number corresponding to the next less logarithm and set it aside. Subtract this less logarithm from the given logarithm, and annex to the remainder as many zeros as may be necessary, and divide this result by the corresponding number taken from the column marked D, continuing the division as long as desirable: annex the quotient to the number set aside. Point off, from the left hand, as many integer figures as there are units in the characteristic of the given logarithm increased by 1; the result is the required number.

If the characteristic is negative, the number will be entirely decimal, and the number of zeros to be placed at the left of the number found from the table, will be equal to the number of units in the characteristic diminished by 1.

This rule, like its converse, is founded on the supposition that the difference of the logarithms is proportional to the difference of their numbers within narrow limits.

EXAMPLE.

1. Find the number corresponding to the logarithm 3.233568.

The decimal part of the given logarithm is .233568 The next less logarithm of the table is

and its corresponding number 1712.

Their difference is

Tabular difference

Hence, the number sought

.233504,

64

253)6400000(25

1712.25.

The number corresponding to the logarithm 3.233568 is .00171225.

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

3. What is the number corresponding to the logarithm

MULTIPLICATION BY LOGARITHMS.

11. When it is required to multiply numbers by means of their logarithms, we first find from the table the logarithms of the numbers to be multiplied; we next add these logarithms together, and their sum is the logarithm of the product of the numbers (Art. 3).

The term sum is to be understood in its algebraic sense; therefore, if any of the logarithms have negative characteristics, the difference between their sum and that of the positive characteristics, is to be taken; the sign of the remainder is that of the greater sum.

EXAMPLES.

1. Multiply 23.14 by 5.062.

log 23.141.364363

log 5.062 0.704322

Product, 117.1347 ... 2.068685

2. Multiply 3.902, 597.16, and 0.0314728 together.

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

3. Multiply 3.586, 2.1046, 0.8372, and 0.0294 together. log 3.586 0.554610

In this example the 2, carried from the decimal part,