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CHAPTER XX.

LOGARITHMS.

394. The logarithm of a number is the exponent of the power to which a constant number must be raised in order to be equal to the proposed number. The constant number is called the base of the system.

Thus, if a denote any positive number except unity, and a2=m, then 2 is the exponent of the power to which a must be raised to equal m; that is, 2 is the logarithm of m in the system whose base is a. If a*=m, then x is the logarithm of m in the system whose base is a.

395. If we suppose a to remain constant while m assumes in succession every value from zero to infinity, the corresponding values of x will constitute a system of logarithms.

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Since an indefinite number of different values may be attributed to a, it follows that there may be an indefinite number of systems of logarithms. Only two systems, however, have come into general use, viz., that system whose base is 10, called Briggs's system, or the common system of logarithms; and that system whose base is 2.718+, called the Naperian system, or hyperbolic system of logarithms.

Properties of Logarithms in general.

396. The logarithm of the product of two or more numbers is equal to the sum of the logarithms of those numbers.

Let a denote the base of the system; also, let m and n be any two numbers, and x and y their logarithms. Then, by the definition of logarithms, we have

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Multiplying together equations (1) and (2) member by mem

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Therefore, according to the definition of logarithms, x+y is the logarithm of mn, since it is the exponent of that power of the base which is equal to mn.

For convenience, we will use log. to denote logarithm, and we have

x+y=log. mn=log. m+log. n.

Hence we see that if it is required to multiply two or more numbers together, we have only to take their logarithms from a table and add them together; then find the number corresponding to the resulting logarithm, and it will be the product required.

397. The logarithm of the quotient of two numbers is equal to the logarithm of the dividend diminished by that of the divisor. If we divide Eq. (1) by Eq. (2), member by member, we shall have

of

ax-y=

m
n

Therefore, according to the definition, x-y is the logarithm

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since it is the exponent of that power of the base a which

m

is equal to That is,

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x—y=log. (1)=log. m—log. n.

Hence we see that if we wish to divide one number by another, we have only to take their logarithms from the table and subtract the logarithm of the divisor from that of the dividend; then find the number corresponding to the resulting logarithm, and it will be the quotient required.

398. The logarithm of any power of a number is equal to the logarithm of that number multiplied by the exponent of the power. If we raise both members of Eq. (1) to any power denoted by p, we have

apxmp.

Therefore, according to the definition, pa is the logarithm of m2, since it is the exponent of that power of the base which is equal to me. That is,

px=log. (mr)=p log. m.

Therefore, to involve a given number to any power, we multiply the logarithm of the number by the exponent of the power; the product is the logarithm of the required power.

399. The logarithm of any root of a number is equal to the logarithm of that number divided by the index of the root.

If we extract the rth root of both members of Eq. (1), we

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Therefore, according to the definition, is the logarithm of Vm. That is

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Therefore, to extract any root of a number, we divide the logarithm of the number by the index of the root; the quotient is the logarithm of the required root.

400. The following examples will show the application of the preceding principles:

Ex. 1. log. (abcd)=log. a+log. b+log. c+log. d.

Ex. 2. log. (abc)=log. a+log. 6+log. c—log. d—log. e.

Ex. 3. log. (amb cr)=m log. a+n log. b+p log. c.

Ex. 4. log.

log. (amba)

=m log. a+n log. b—p log. c.

Ex. 5. log. √ab=(log. a+log. b).

Ex. 6. log.

3/ab2c4

d5

-={log. a+2 log. b+4 log. c—5 log. d}.

Ex. 7. log. (a3 √a3)=log. (a) — 15 log. a.

Ex. 8. log.(a2x2)=log. {(a+x) (a-x)}=log. (a+x)+log.(a−x).

Ex. 9. log. Va2x2= log. (a+x)+1 log. (a−x).

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401. In all systems of logarithms, the logarithm of unity is zero.

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if we make n=1, the corresponding value of x will be 0, since ao=1, Art. 75; that is, log. 1=0.

402. In all systems of logarithms, the logarithm of the base is

unity. For that is,

a1=a; log. a=1.

Common Logarithms.

403. Since the base of the common system of logarithms is 10, all numbers in this system are to be regarded as powers of 10. Thus, since

10°=1,
101=10,

we have

log. 1=0;

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From this it appears that in Briggs's system the logarithm any number between 1 and 10 is some number between 0 and 1; that is, it is a fraction less than unity, and is generally expressed as a decimal. The logarithm of any number between 10 and 100 is some number between 1 and 2; that is, it is equal to 1 plus a decimal. The logarithm of any number between 100 and 1000 is some number between 2 and 3; that is, it is equal to 2 plus a decimal; and so on.

404. The same principle may be extended to fractions by means of negative exponents. Thus, since

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Hence it appears that the logarithm of every number between 1 and 0.1 is some number between 0 and -1, or may be represented by -1 plus a decimal. The logarithm of every number between 0.1 and 0.01 is some number between -1 and

-2, or may be represented by -2 plus a decimal. The logarithm of every number between 0.01 and 0.001 is some number between 2 and 3, or may be represented by -3 plus a decimal, and so on.

405. Hence we see that the logarithms of most numbers must consist of two parts, an integral part and a decimal part. The former part is called the characteristic or index of the logarithm. The characteristic may always be determined by the following

RULE.

The characteristic of the logarithm of any number is equal to the number of places by which the first significant figure of that number is removed from the unit's place, and is positive when this figure is to the left, negative when it is to the right, and zero when it is in the unit's place.

Thus the characteristic of the logarithm of 397 is +2, and that of 5673 is +3, while the characteristic of the logarithm of 0.0046 is -3.

406. The same decimal part is common to the logarithms of all numbers composed of the same significant figures.

For, since the logarithm of 10 is 1, it follows from Art. 397 that if a number be divided by 10, its logarithm will be diminished by 1, the decimal part remaining unchanged. Thus, if we denote the decimal part of the logarithm of 3456 by m, we shall have

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407. The table on pages 290, 291, contains the decimal part of the common logarithm of the series of natural numbers from 100 to 999, carried to four decimal places. Since these numbers are all decimals, the decimal point is omitted, and the characteristic is to be supplied according to the rule in Art. 405.

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