Imágenes de páginas
PDF
EPUB
[blocks in formation]

then x is the index of the power to which a must be raised in order to equal N.

For some purposes, this idea is presented in these words: If a= N, then x is the logarithm of N to the base a.

The latter statement is taken as the definition of a logarithm, and is expressed by mathematical symbols in this manner, viz.:

[blocks in formation]

Equations (1), (2), are equivalent; they are merely two different ways of stating a certain connection between the three quantities a, x, N. For example, the relations

238, 5625, 10-100.001,

may also be expressed by the equivalent logarithmic equations, log, 83, log, 625 = 4, log10 .001 - 3.

EXAMPLES.

1. Express the following equations in a logarithmic form :

38 = 27, 44 = 256, 112 = 121, 93 = 728, 78 =

343, mb = p.

2. Express the following equations in the exponential form:

log2 83, log5 625 = 4, log10 1000 = 3, log2 64 = 8, logn P = a.

3. When the base is 2, what are the logarithms of 1, 2, 4, 8, 16, 32, 64, 128, 256 ?

4. When the base is 5, what are the logarithms of 1, 5, 25, 125, 625, 3125? 5. When the base is 10, what are the logarithms of 1, 10, 100, 1000, 10,000, 100,000, 1,000,000, .1, .01, .001, .0001, .00001, .000001 ?

6. When the base is 4, and the logarithms are 0, 1, 2, 3, 4, 5, what are the numbers?

7. When the base is 10, between what whole numbers do the logarithms of the following numbers lie: 8, 72, 235, 1140, 3470, .7, .04, .0035 ?

3. Properties of logarithms. Since a logarithm is the index of a power, it follows that the properties of logarithms must be derivable from the properties of indices; that is, from the laws

of indices. The laws of indices are as follows (a, m, n, being any finite quantities):

[merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small]

MN= am+"; whence, loga MN = m + n = loga M + loga N. (3) [If P= a", then log, P=p, MNP = a+n+p;

whence, log, MNP=m+n+p=loga M+loga N+ loga P.]

[blocks in formation]

mr

Also, M=(a)" am"; whence, loga M" = rm = r loga M. (5)

Also,

whence,

=

[merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small]

The results (3)-(6) state the properties, or are the laws of logarithms. They may be expressed in words as follows:

(1) The logarithm of the product of any number of factors is equal to the sum of the logarithms of the factors.

(2) The logarithm of the quotient of two numbers is equal to the logarithm of the numerator diminished by the logarithm of the denominator.

(3) The logarithm of the rth power of a number is equal to r times the logarithm of the number.

(4) The logarithm of the rth root of a number is equal to th of the logarithm of the number.

Hence, if the logarithms (i.e. the exponents of powers) of numbers be used instead of the numbers themselves, then the operations of multiplication and division are replaced by those of addition and subtraction, and the operations of raising to powers and extracting roots, by those of multiplication and division.

4. Common system of logarithms. Any positive number except 1 may be chosen as the base; and to the base chosen there corresponds a set or system of logarithms. In the common or decimal system the base is 10, and, as will presently appear, this system is a very convenient one for ordinary numerical calculations.* In what follows, the base 10 is not expressed, but it is always understood that 10 is the base. The logarithm of a number in the common system is the answer to the question: "What power of 10 is the number ?"

Since

1=10°, 10= 101, 100=102,

it follows that

1000=103,

10000=101, ...,

log 1= 0, log 10= 1, log 100= 2, log 1000= 3, log 10000= 4,

This also shows that the logarithms of numbers

between 1 and

10 lie between 0 and 1,

between 10 and 100 lie between 1 and 2,

between 100 and 1000 lie between 2 and 3, and so on.

(1)

....

For example,

9=10-95424,

247=

102.39270,

1453=103.16227 ;

or log 9.95424, log 247 2.39270, log 1453 = 3.16227.

=

Most logarithms are incommensurable numbers. (See Art. 9.) The decimal part of the logarithm is called the mantissa, the

*The base of the natural system of logarithms is an incommensurable number, which is always denoted by the letter e and is approximately equal to 2.7182818284.

integral part of the logarithm is called the index or charac teristic.

The two great advantages of the common system, as will now be shown, are:

(1) The characteristic of a logarithm can be written on mere inspection;

(2) The position of the decimal point in a number affects the characteristic alone, the mantissa being always the same for the same sequence of figures.

Since .1 == 10-1,

10

.01 = Too = 10-2,

τόσ

.001010-3, .0001 = 10000 = 10−4, ....*,

it follows that

log.11, log .012, log .0013, log .0001-4, etc. (2)

From (1) and (2) comes the following rule for finding the characteristic:

When the number is greater than 1, the characteristic is positive and is one less than the number of digits to the left of the decimal point; when the number is less than 1, the characteristic is negative, and is one more than the number of zeros between the decimal point and the first significant figure.

When a change is made in the position of the decimal point in a number, the value of the number is changed by some integral power of 10. Its logarithm is then changed by a whole number only, and, consequently, its mantissa is not affected. For example, 2538000 = 2538 × 103;

25.38 = 2538 × 10−2,

and hence, log 25.38 = log 2538-2, log 2538000= log 2538 +3.

Accordingly, it is necessary to put only the mantissas of sequences of integers in the tables.

5. Negative characteristics. In common logarithms the mantissa is always kept positive. Thus, for example, log 25380 = 4.40449; log .002538 log 1000000 = 3.40449 — 6

=

log

2538 1000000

=

log 2538

[blocks in formation]

This logarithm is usually written 3.40449, in order to show that the minus sign affects the characteristic alone. In order to avoid the use of negative characteristics, 10 is often added to the logarithm and 10 placed after it.

Thus 3.40449 is written 7.40449 -- 10.

The second form is more convenient for purposes of calculation. Special care is necessary in dealing with logarithms because of the fact that the mantissa is always positive, while the characteristic may be either positive or negative. Some typical examples involving negative characteristics are given below.

[blocks in formation]

A result like (1) is always put in the form (2), in which the number placed after the logarithm is -10.

Ex. 3 may also be worked thus:

(-1.83471) x 2-2 +1.66942 1.66942.

=

4. Division. 3.27412 ÷ 4 = (37.27412 - 40) ÷ 4 = 9.31853 - 10.

=

As in Ex. 3 care is taken that, finally, the number after the logarithm be- 10.

5. 2.34175 ÷ 5 = (48.34175 – 50) ÷ 5 = 9.66835 — 10.

-

[merged small][merged small][merged small][ocr errors][merged small][merged small]

The method of finding the logarithms in the tables when the numbers are given, and the way to find the numbers when the logarithms are given, are usually explained in connection with the tables of logarithms.

6. Exercises in logarithmic computation. On looking at the laws of logarithms, (3)-(6), Art. 3, it is apparent that logarithms cannot assist in the operations of addition and subtraction. Logarithms are of no service in computing expressions of the forms

« AnteriorContinuar »