We have, observing, the rule of division, to subtract the exponent of the divisor from that of the dividend in order to obtain that of the quotient.

Since 1

l' is the exponent of the power to which it is necessary to raise a the base, in order to produce it follows

that — l' is the logarithm of 2

n'

n

n

i. e. the logarithms of the

quotient is equal to the difference between the logarithms of the divisor and dividend.

Before explaining other operations by means of logarithms, we shall exhibit some principles derived from those just demonstrated.

50. The base of the common system being 10, the common logarithm of 10 is 1. (art. 44.) Hence if any number be multiplied or divided by any number of times 10, the logarithm of the result will be equal to the logarithm of the given number increased or diminished by the same number of times 1. This 1 being an entire number, the decimal part of the logarithm of the given number will not be altered by this addition or diminution, but only the characteristic.

Thus 397940 which is the decimal part of the logarithm of 2500, is also of 25000, and of 250000, or of 250, or of 25. The characteristics belonging to these different numbers are different. That of the log. of 2500 is 3; that of the log. of 25000 is 4; that of the log. of 25 is 1.

Any number is divided by a multiple of 10, by pointing off from the right as many places for decimals, as the divisor is times 10.

Thus 2348 divided by 10, by twice 10, by three times 10, becomes successively 234.8 23.48 2.348. The decimal part of the logarithms of these last three numbers, will be the same, the characteristic being one less each time that we divide by 10 or remove the decimal point one place to the left. The characteristic of the first, which is between 100= 102 and 1000=103,is 2. The characteristic of the second is 1; and the characteristic of the last is 0, since 2.348 is less than 10, or 101.

The decimal part of the logarithm of a number consisting of significant figures, either followed or preceded by ciphers, will be the same as if the ciphers were absent. Thus the decimal part of the logarithm of 482000 or of .00482 is the same as the decimal part of the logarithm of 482.

The following table illustrates the theory of the characteristic.

The characteristic of the log. of 482000 is 5

From the above, it appears that the characteristic of the logarithm of a decimal fraction is negative; the decimal part of the same logarithm is, however, positive. The actual value of the whole logarithm will be therefore a negative quantity somewhat less than the characteristic. That the logarithms of proper fractions ought to be negative, appears from the fact, that since a fraction expresses the quotient of the numerator divided by the denominator, applying the rule

for division by logarithms, the greater logarithm would have to be subtracted from the lesser and the remainder would of course be negative.

From the above principles are derived the following rules:

1. To find the logarithm of a number consisting of significant figures with any number of ciphers annexed, find the logarithm of the significant figures, and make the characteristic one less than the number of figures in the given number including the ciphers.

2. To find the logarithm of a decimal or mixed number, consider the number as entire; find the decimal part of its logarithm, and make the characteristic one less than the number of figures in the entire part of the given number.

4. To find the logarithm of a decimal number having ciphers at the left; look for the logarithm of the significant figures, and make the characteristic negative* and one more than the number of ciphers at the left of the given decimal.

51. We proceed now to the method of determining the logarithm of a number beyond the limits of the table. This method is by a simple calculation from the logarithms of numbers which the table contains, and depends upon the fact that the difference of any two numbers bears the same proportion to the difference of their logarithms, that the difference of two other numbers does to the difference of their logarithms, which is nearly true.

Take two numbers in the table differing from each other by 100 as the numbers 843700 and 843800 and a third number 843742 differing from the first of these by 42. The loga

* It is customary to write the negative sign over the characteristic, thus, 2.175 It affects the characteristic alone and not the decimal part of the logarithm.

rithm of the first number 843700 is given by the tables and is 5.926188

The logarithm of the second number 843800 is 5.926240

Thier difference is

*52

which may be found by subtraction, but to save this trouble the subtraction is performed and the difference is written in the column marked D, the last of each page in the table. Then

adding this to the logarithm of 843700 which is 5.926188

the sum, rejecting the last two places 42 which go

beyond the usual number is

which is the logarithm of 843742.

.00002142

5.926209

Had the first two numbers differed by 1000 instead of 100 the divisor in the value of x would have been 1000 and the quotient would have extended three places beyond the usual. The inaccuracy of this method increases with the number of additional figures beyond four, in the number the logarithm of which is to be found.

From the above process may be observed the following rule: To find the logarithm of a number beyond the limits of the table. Enter the table with the first four figures of the given number, and find the corresponding logarithm. From the column marked n take out the number opposite to this logarithm, and multiply it by the remaining figures of the proposed number, reject from the product as many figures to the right as there are in the multiplier, and add the rest of the product to the logarithm already found.

The remainder is 52, but if the decimals had been carried beyond six places 'n the tables, it would have been 51..

EXAMPLE I.

Required the logarithm of 739245.

The decimal part of the log. of 7392 is 868762. the number in column D is

Multiplying this by the remaining figures of the given number

59

45

295

236

2655

Product,

From this product reject as many figures to the right as are contained in the multiplier, that is two in this case, and add the rest to the logarithm before found, namely

The sum is

868762

868789*

which is the decimal part of the log. of 739245 required. Prefixing the proper characteristic, we have 5.868789.

52. By referring to the proportion of art. 51, and putting

the value of x for the fourth term we have