If the above principles be extended to other numbers, it will appear, that the logarithm of any number, not an exact power of ten, is made up of two parts, an entire and a decimal part. The entire part is called the characteristic of the logarithm, and is always one less than the number of places of figures in the given number.

3. The principal use of logarithms, is to abridge numerical computations.

Let denote any number, and let its logarithm be denoted by m; also let N denote a second number whose logarithm is n; then, from the definition, we shall have, 10" = N (2).

10m = M (1)

Multiplying equations (1) and (2), member by member, we have,

10m+"=MXN or, m+n=log (MXN); hence,

The sum of the logarithms of any two numbers is equal to the logarithm of their product.

4. Dividing equation (1) by equation (2), member by member, we have,

The logarithm of the quotient of two numbers, is equal to the logarithm of the dividend diminished by the logarithm of the divisor.

5. Since the logarithm of 10 is 1, the logarithm of the product of any number by 10, will be greater by 1 than the logarithm of that number; also, the logarithm of the quotient of any number divided by 10, will be less by 1 than the logarithm of that number.

Similarly, it may be shown that if any number be mul tiplied by one hundred, the logarithm of the product will be greater by 2 than the logarithm of that number; and if any number be divided by one hundred, the logarithm of the quotient will be less by 2 than the logarithm of that number, and so on.

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 thein on the same horizontal line. The last column on each page, headed D, shows the difference between the loga This difference is

rithms of two consecutive numbers. 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 differences of any two consecutive logarithms on this line.

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, ◊ must be written for each dot, and the two figures to be prefixed are, as before, taken from the line below: Thus,

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.

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,
:

hence, 56.55 is the number to be added to the logarithm before found.

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 supposi tion 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 1.846741? Ans. .702653.