Thus by means of the powers of a, the exponents of which are negative entire or fractional, we may produce all possible positive numbers less than 1.

If on the other hand we suppose a less than unity, still all possible positive numbers may be produced by means of the different powers of a, only the order in which they are produced will be reversed.

We see therefore, that all possible positive numbers may be produced by means of any positive number whatever a, different from unity, by raising this number to the requisite powers.

It is necessary, that a should be different from unity, otherwise the same number will be produced, whatever value we assign to x.

197. Let it now be supposed that we have made a table containing in one column all entire numbers, and by the side of these in another column the exponents of the powers, to which it is necessary to raise a constant number in order to produce these numbers; this would be a table of logarithms.

The logarithm of a number, is, therefore, the exponent of the power, to which it is necessary to raise a given or invariable number, in order to produce the proposed number.

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Thus in the equation a y, x is the logarithm of y; in like manner in the equation 2o 64, 6 is the logarithm of 64. The logarithm of a number is indicated by writing before it the first three letters of the word logarithm, or more simply by placing before it the letter L.

The invariable number, from which the others are formed is called the base of the table. It may be taken at pleasure either greater or less than unity, but should remain the same for the formation of all numbers, that belong to the same table.

Since a° - 1, and a1: =a, whatever number may be assumed for the base of the table, the logarithm of the base will be unity and the logarithm of unity will be 0.

198. We proceed to show the properties of logarithms in relation to numerical calculations.

1. Let there be the series of numbers y, y', y', .... to be multiplied together. Let a represent the base of a system of logarithms, which we suppose already calculated, and let x, x', ".. be the logarithms of y, y', y', ...; by the definition of a logarithm we have

y=a", y' = a*, y′′ = a*";

multiplying these equations member by member, we have yy'y" = a=+=+=",

whence log yy'y"=x+x′+x"= log y+logy' + log y". That is, the logarithm of a product is equal to the sum of the logarithms of the factors of this product.

If then a multiplication be proposed, we take from a table of logarithms the logarithms of the numbers to be multiplied; the sum of these logarithms will be the logarithm of the product sought. Seeking therefore this logarithm in the table, the number corresponding to it will be the product sought. Thus by means of a table of logarithms addition may be made to take the place of multiplication.

2. Let it be required to divide the number y by the number y'; let x, x' be the logarithms of these numbers, we have the equations

That is, the logarithm of a quotient is equal to the difference between the logarithm of the divisor and that of the dividend.

If then it be proposed to divide one number by another, from the logarithm of the dividend we subtract the logarithm of the divisor, the result will be the logarithm of the quotient; seeking therefore this logarithm in the tables the number corresponding will be the quotient sought. Thus, by means of a table of logarithms, subtraction may be made to take the place of division.

3. Let it next be required to raise the number y denoted by m, we have the equation

a = y;

raising both members to the mth power, we have

whence the logarithm of y =mx=m log y.

That is, the logarithm of any power of a number is equal to the product of the logarithm of this number by the exponent of the power.

To form any power whatever of a number by means of a table of logarithms, we multiply, therefore, the logarithm of the proposed number by the exponent of the power, to which it is to be raised; the number in the table corresponding to this product, will be the power sought.

4. Again, let it be required to find the nth root of y. We have as before a = - y;

whence taking the nth root of both members, we have

That is, the logarithm of the root of any degree whatever of a number is equal to the logarithm of this number divided by the index of the root.

Thus by the aid of a table of logarithms a number may be raised to a power by a simple multiplication, and its root may be extracted by a simple division.

FORMATION OF TABLES.

199. The properties of logarithms demonstrated above are altogether independent of the number a or their base. We may therefore form an infinite variety of tables of logarithms by putting for a all possible numbers except unity.

If it be required to construct a table of logarithms the base of which is 2, in the equation 2=y, we make y equal successively to the numbers 1, 2, 3 and determine by the methods

explained, art. 195, the values of x corresponding.

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We thus obtain the values of x exactly, if y be a perfect power of 2, or otherwise with such degree of approximation as we please

To calculate the logarithm of 3, for example, we have the equation 23, from which we deduce

Whence stopping at the fourth integrant fraction, and forming

19

12'

the reduction corresponding, we have x= or reducing this last to a decimal we have x=1.583 accurate to the third deci

mal figure,

200. In the calculation of a table of logarithms, it will be sufficient to calculate directly the logarithms of the prime numbers 1, 2, 3, 5..., the logarithms of compound numbers may then be obtained by adding the logarithms of the prime factors, which enter into them. To find the logarithm of 35, for example, we have 35=5X7; whence log 35=log 5+ log7; having already calculated the logarithms of 5 and 7, the logarithm of 35 will be found therefore by adding the logarithm of 5 to that of 7.

Since moreover the logarithm of a fraction will be equal to the logarithm of the numerator minus the logarithm of the denominator, it will be sufficient to place in the tables the logarithms of entire numbers.

201. Below we have a table of logarithms of numbers from 1 to 30 inclusive, the base of the system is 2, and the logarithms are calculated to 4 places of decimals.

202. The most convenient number for a base to a system of logarithms, and the one employed in the construction of the tables in common use is 10.

If in the equation 10y we make successively

Therefore in a table of logarithms, the base of which is 10, 1o. the logarithms of numbers greater than unity are positive and go on increasing from 0 to infinity. 2°. The logarithms of numbers less than unity are negative, and their absolute values are so much the greater as the fractions are smaller; whence if we take a fraction less than any assignable quantity, the logarithm of this fraction will be negative, and its absolute value will be greater than any assignable quantity. On this account we say that the logarithm of 0 is an infinite negative quantity. 3o. The logarithms of all numbers below 10 are fractions; the logarithms of numbers between 10 and 100 are 1 and a fraction; the logarithms of numbers between 100 and 1000 are 2 and a fraction; those of numbers between 1000 and 10000 are 3 and a fraction; and in general, the whole number which precedes the fraction in the logarithm is less by one than the number of figures in the number corresponding to the logarithm. On this account it is called the index or characteristic of the logarithm, since it serves to indicate the order of units, to which the number corresponding to the logarithm belongs. Thus in the logarithm 3.75527 the characteristic 3 shows that the number corresponding to this logarithm consists of 4 figures or is comprised between 1000 and 10000.

203. The logarithm of a number being given, the logarithm of a number 10, 100, times greater is found by adding 1, 2, . . . units to the characteristic only; indeed log

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(y × 10") = log y + log 10" log y +n;

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