It is seen that each number in (3) is a power of the fixed number 2, and that the corresponding number in (1) is the exponent of that power.

It is thus apparent that these products are found by adding the exponents of the fixed number 2, and taking the corresponding power of 2 from (3).

Show that quotients may be found by subtracting the exponents of 2 and taking the corresponding power of 2 from (3).

We now come to the consideration of a scheme by which the powers of any number may be arranged in a table and products, quotients, powers, and roots be found by the addition, subtraction, multiplication, and division of exponents.

463. The exponent of the power to which a fixed number must be raised to produce a given number is called the Logarithm of that number.

When 10 is the fixed number, and 100 the given number, the exponent 2 is the logarithm, because 102 = 100; if 1000 were the given number, its logarithm would be 3, because 103 = 1000. Since 10 10,000, what is the logarithm of 10,000?

464. The fixed number whose exponent is the logarithm of a given number is called the Base. Any positive number except 1 may be taken as a base for computing logarithms. The number one is excepted because its powers are no greater than itself.

465. When the base is 10, the logarithms are called Common or Briggs's Logarithms. These are the only ones used for numerical computations. If no base is mentioned, the base 10 is understood. In this system all numbers are regarded as powers of ten.

In these expressions 10 is the base, and the exponents are the logarithms. Thus, 2 is the logarithm of 100 to the base 10, etc. Using the logarithmic form, we have

log 10 1002, or simply log 100=2;

log 20 1.3010+;

=

log 124 = 2.0934 +.

Although the exponent indicating the power to which 10 must be raised cannot be expressed exactly when it is not an integral power of 10, it can be found to any required degree of accuracy by using approximate values. The laws which were found to apply to rational exponents apply also to irrational exponents.

466. In general, when a is the base, a the exponent, and n the given number, we have an, in which a is the logarithm of n to the base a. It is written log。 n = x.

In theoretical investigation use is made of what is known as Natural Logarithms. The base of this system is 2.7182818 ....., denoted by the letter e. When 10 is the base it is not usually

expressed.

Logarithms were invented about the year 1614 by a Scotchman, John Napier. The base 10 was suggested by an Englishman, Henry Briggs, a contemporary of Napier. The chief advantage of the base 10 lies in the fact that it is also the base of our decimal system, and that as a consequence from the known logarithm of a given number there may be derived by inspection the logarithms of all numbers differing from the given number only in the position of the decimal point.

468. It is therefore evident that the logarithm of any number between 1 and 10 is greater than 0 and less than 1, that is, it is 0 plus a fraction; that the logarithm of any number between 10 and 100 is greater than 1 and less than 2, that is, it is 1 plus a fraction; and so on. Thus, the logarithm of 9 has been found to be 0.9542 +, and that of 25 to be 1.3979 +.

It is also evident that the logarithm of a number greater than 1 is positive, while that of a number less than 1 and greater than 0 is negative.

469. Only exact powers of 10 have integral logarithms. All other numbers have logarithms that cannot be expressed exactly. They are 0 or an integer, plus a decimal that does not terminate.

Only the numbers between 1 and 10 have logarithms that are wholly fractional that are 0 plus a decimal.

470. The decimal part of a logarithm is called the Mantissa; the integral part is called the Characteristic.

(a) log 16 = 1.2041; (b) log 100 = 2; (c) log 4 = 0.6021.

In (a), .2041 is the mantissa, 1 the characteristic; in (b), there is no mantissa - the logarithm is integral, 100 being a power of 10; in (c), the characteristic is 0.

Why?

471. Since 6043 lies between 103 and 101, that is, between 1000 and 10,000, its logarithm lies between 3 and 4. By calculation it has been found to be 3.7812, the integral part, 3, being the characteristic, and the decimal part, .7812, the mantissa.

It is evident from these examples that:

1. If a number is greater than 1, the characteristic of its logarithm is positive and is 1 less than the number of integral places in the number.

Thus, if a characteristic is 3, we know at once that the number contains 4 integral places.

2. If a positive number is less than 1, the characteristic of its logarithm is negative, and is numerically 1 greater than the number of ciphers between the decimal point and the first significant figure. When the minus sign is written over the characteristic, it denotes that it alone is negative.

Thus, if a characteristic is -3, we know at once that the decimal has 2 ciphers following the decimal point. The mantissa is always positive.

3. If numbers are expressed by the same figures in the same order, their logarithms differ only in their characteristics, the decimal parts remaining the same.

This follows from the fact that moving the decimal point is equivalent to dividing or multiplying the number by a power of 10, which increases or diminishes the logarithm by the exponent of that power; and this exponent, being integral, affects only the characteristic.

The foregoing explains why tables of logarithms contain only the mantissas. The characteristic can always be found by inspection.

472. 1. What is the characteristic of the logarithm of a number of 2 integral places? Of 3? Of 5? Of 10?

2. What is the characteristic of the logarithm of .2? Of .05? Of .007 ?

3. If the logarithm of 6742 is 3.8288, what is the logarithm of 67,420? Of 674,200? Of 674.2? Of 67.42 ?

Since only positive numbers are employed as bases in any system of logarithms (Art. 464), negative numbers are not considered as having logarithms, but operations involving negative numbers are performed as if the numbers were positive, and the proper sign prefixed to the result.

473. A table of four-place logarithms is given on pages 406– 407. This table contains the mantissas of the logarithms of all numbers under 1000, the characteristic and decimal point. being omitted. The given numbers from 10 to 99 are found in the columns headed "N"; the mantissas are found in the other columns.

In this table the mantissas are correct to three decimal places. The fourth figure is also correct unless the fifth is 5 or more, in which case the fourth is increased by 1, just as we write $1.38 for $1.375. Four-place tables are sufficiently exact for all ordinary calculations; for work requiring greater accuracy more extended tables are used.

474. To find the Logarithm of a Number.

The mantissa of the logarithm of any integral number from 10 to 100 can be found opposite the given number on pages 406-407. The mantissas of the numbers 1, 2, 3, ..., 9, are the same as those of 10, 20, 30, ..., 90 (Art. 471, 3), so that the table may be conveniently begun with the number 10. The following examples will illustrate the method of finding the logarithms of any other numbers.

1. Find the logarithm of 678, of 67.8, of .678, and of .0678.

On the second page of the table, the first two figures of 678 are found in the left-hand column headed “N." On a line with these and in the column headed by the third figure 8, we find 8312. With the decimal point prefixed, the result .8312 is the mantissa of the log 678.