CHAPTER XXII.

LOGARITHMS.

399. In the common system of notation the expression of numbers is founded on their relation to ten.

Thus, 3854 indicates that this number contains 103 three times, 102 eight times, 10 five times, and four units.

400. In this system a number is represented by a series of different powers of 10, the exponent of each power being integral. But, by employing fractional exponents, any number may be represented (approximately) as a single power

of 10.

401. When numbers are referred in this way to 10, the exponents of the powers corresponding to them are called their logarithms to the base 10.

For brevity the word "logarithm" is written log.

It is evident that the logarithms of all numbers between

402. The fractional part of a logarithm cannot be expressed exactly either by common or by decimal fractions; but decimals may be obtained for these fractional parts, true to as many places as may be desired.

If, for instance, the logarithm of 2 be required; log 2 may be supposed to be .

Then 10 2; or, by raising both sides to the third power, 10 = 8, a result which shows that is too large.

Since is too large and too small, log 2 lies between 1 and 1; that is, between .33333 and .30000.

In supposing log 2 to be, the error of the result is 1058 = 2.2. In supposing log 2 to be, the error of the result is 1000-1024 F0200 -.024; log 2, therefore, is nearer to than to

1000

The difference between the errors is .2-(-.024) = .224, and the difference between the supposed logarithms is .33333 - .3 .03333.

The last error, therefore, in the supposed logarithm may be considered to be approximately of .03333 = .0035 nearly, and this added to .3000 gives .3035, a result a little too large.

By shorter methods of higher mathematics, the logarithm of 2 is known to be 0.3010300, true to the seventh place.

403. The logarithm of a number consists of two parts, an integral part and a fractional part.

Thus, log 2 = 0.30103, in which the integral part is 0, and the fractional part is .30103; log 20 = 1.30103, in which the integral part is 1, and the fractional part is .30103.

404. The integral part of a logarithm is called the characteristic; and the fractional part is called the mantissa.

405. The mantissa is always made plus. Hence, in the case of numbers less than 1 whose logarithms are minus, the logarithm is made to consist of a minus characteristic and a plus mantissa.

406. When a logarithm consists of a minus characteristic and a plus mantissa, it is usual to write the minus sign over the characteristic, or else to add 10 to the characteristic and to indicate the subtraction of 10 from the resulting logarithm.

Thus, log .2 = 1.30103, and this may be written 9.30103 - 10.

407. The characteristic of a logarithm of an integral number, or of a mixed number, is one less than the number of integral digits.

Thus, from 401, log 1 = 0, log 10 = 1, log 100 = 2. Hence, the logarithms of all numbers from 1 to 10 (that is, of all numbers consisting of one integral digit), will have 0 for characteristic; and the logarithms of all numbers from 10 to 100 (that is, of all numbers consisting of two integral digits), will have 1 for characteristic; and so on, the characteristic increasing by 1 for each increase in the number of digits, and therefore always being 1 less than that number.

408. The characteristic of a logarithm of a decimal fraction is minus, and is equal to the number of the place occupied by the first significant figure of the decimal.

Thus, from 401, log .1=1, log .01 = -2, log .001 -3. Hence, the logarithms of all numbers from .1 to 1 will have 1 for a characteristic (the mantissa being plus); the logarithms of all numbers from .01 to .1 will have - 2 for a characteristic; the logarithms of all numbers from .01 to .001 will have 3 for a characteristic; and so on, the characteristic always being minus and equal to the number of the place occupied by the first significant figure of the decimal.

409. The mantissa of a logarithm of any integral number or decimal fraction depends only upon the digits of the number, and is unchanged so long as the sequence of the digits remains the same.

For, changing the position of the decimal point in a number is equivalent to multiplying or dividing the number by a power of 10. Its logarithm, therefore, will be increased or diminished by the exponent of that power of 10; and, since this exponent is integral, the mantissa of the logarithm will be unaffected.

410. The advantage of using the number 10 as the base of a system of logarithms consists in the fact that the mantissa depends only on the sequence of digits, and the characteristic on the position of the decimal point.

411. As logarithms are simply exponents therefore (§ 148), The logarithm of a product is the sum of the logarithms of the factors.

=

=

= 0.3010+ 1.0000 1.3010; log 2000 = log (2 × 1000) = log 2 + log 1000, = 0.3010+ 3.0000 = 3.3010;

log .2log (2.1)= log 2+ log .1,

= 0.3010+ 9.000 10 9.3010 - 10;

log .02 = log (2 x .01) = log 2 + log .01,

0.30108.000010 8.3010 - 10.

EXERCISE LXXX.

=

Given: log 20.3010; log 3=0.4771; log 5=0.6990; log 7=0.8451.

Find the logarithms of the following numbers by resolv

ing the numbers into factors, and taking the sum of the logarithms of the factors:

The logarithm of a power of a number is equal to the logarithm of the number multiplied by the exponent of the

413. As logarithms are simply exponents, therefore (§ 381),

The logarithm of a root of a number is equal to the logarithm of the number multiplied by the index of the root.

for, since 20-20=0, the addition of 20 to the 7, and of 20 to the - 10, produces no change in the value of the logarithm

414. In simplifying the logarithm of a root the equal plus and minus numbers to be added to the logarithm must be such that the resulting minus number, when divided by the denominator of the index of the root, shall give a quotient of -10.