We must not, however, suppose that the number 3.4727564 is the exact log arithm of 2970, or that

2970 (10)+2
3.4727561

accurately. The above is only an approximate value of the logarithm of 2970 we can obtain the exact logarithm of very few numbers, but taking a sufficient number of decimals we can approach as nearly as we please to the true logarithm, as will be seen when we come to treat of the construction of tables.

198. It has been shown that in Briggs' system the logarithm of 1 is 0, consequently, if we wish to extend the application of logarithms to fractions, we must establish a convention by which the logarithms of numbers less than 1 may be represented by numbers less than zero, i. e. by negative numbers. Extending, therefore, the above principles to negative exponents, since

It appears, then, from this convention, that the logarithm of every number between 1 and .1, is some number between 0 and 1; the logarithm of every number between .1 and .01, is some number between 1 and -2; the logarithm of every number between .01 and .001, is some number between - 2 and -3; and so on.

From this will be understood the rule given in books of tables, for finding the characteristic or index of the logarithm of a decimal fraction, viz. The index of any decimal fraction is a negative number, equal to unity, added to the number of zeros immediately following the decimal point. Thus, in searching for a logarithm of the number such as .00462, we find in the tables opposite to 462 the number 6646420; but since .00462 is a number between .001 and .0001, its logarithm must be some number between 3 and 4, i. e. must be 3 plus a fraction, the fractional part is the number 6646420, which we have found in the tables, affixing to this the index ·3, and interposing a decimal point, we have —3. 6646420, the logarithm of .0046?.

General Properties of Logarithms.

199. Let N and N' be any two numbers, ≈ and a their respective logarithms, a the base of the system. Then, by definition,

. by definition, x+x is the logarithm of N N', that is to say,

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

II. Divide equation (1) by (2),

The logarithm of a fraction, or of the quotient of two numbers, is equal to the logarithm of the numerator minus the logarithm of the denominator.

III. Raise both members of equation (1) to the power of n.

N" = anx

... by definition, nx is the logarithm of N", that is to say,

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

IV. Extract the n' root of both members of equation (1).

ΝΤ = a

.. by definition, is the logarithm of N, that is to say,

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

Combining the two last cases, we shall find,

It is of the highest importance to the student to make himself familiar with the application of the above principles to algebraic calculations. The following examples will afford a useful exercise:

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

1o. If a⇒ 1, making x = 0, we have N=1; the hypothesis x=1 gives x = a. As x increases from 0 up to 1, and from 1 up to infinity, N will increase from 1 up to a, and from a up to infinity; so that a being supposed to pass through all intermediate values, according to the law of continuity, N increases also, but with much greater rapidity. If we attribute negative values to x, we have N = a−*, or N=1. Here, as a increases, N diminishes, so that x being supposed to increase negatively, N will decrease from 1 towards 0, the hypothesis x = ∞ gives N = 0.

1

2o. If a 1, put à = , where b1, and we shall then have N = or Nb, according as we attribute positive or negative values to x. We here arrive at the same conclusion as in the former case, with this difference, that when x is positive N1, and when x is negative N > 1.

3o. If a = 1, then N = 1. whatever may be the value of x.

From this it appears, that,

I. In every system of logarithms, the logarithm of 1 is 0, and the logarithm of the base is 1.

II. If the base be1, the logarithms of numbers > 1 are positive, and the logarithms of numbers ▲ 1 are negative. The contrary takes place if the base be 1.

III. The base being fixed, any number has only one real logarithm; but the sume number has manifestly a different logarithm for each value of the base, so that every number has an infinite number of real logarithms, Thus, since y2 = 81, and 3* = 81, 2 and 4 are the logarithms of the same number 81, according as the base is 9 or 3.

IV. Negative numbers have no real logarithms, for attributing to x all values from Co up to +∞, we find that the corresponding values of N are positive numbers only, from 0 up to + ∞.

The formation of a table of logarithms consists in determining and registering the values of x which correspond to N = 1, 2, 3, . . . . in the equation,

the logarithms increase in arithmetical progression, while the numbers increase in geometrical progression; 0 and 1 being the first terms of the corresponding series, and the arbitrary numbers a and m the common difference and the common ratio,

We may, therefore, consider the systems of values of x and y, which satisfy the equation N = ax, as ranged in these two progressions. 201. In order to solve the equation

c = ах

where cand a are given, and where x is unknown, we equate the logarithms of the two members, which gives us

If we have an equation a' = b, where z depends upon an unknown quantity x, and we have

2 = Ax" + Bx11 +

solution of the equation of the n1 degree.

= K some known number, the problem depends upon the

an equation of the second degree, from which we find x2, x = 3

x2

(m log. clog. f) x' (n log. b + p log. f)x + a log. b = 0 a quadratic equation, from which the value of x may be determined.

202. Let N and N+1 be two consecutive numbers, the difference of their logarithms, taken in any system, will be

log. (N+1) - log. N

= log (+1)
(N =

= log. (1)

a quantity which approaches to the logarithm of 1, or zero, in proportion as

1

N

decreases, that is, as N increases. Hence it appears, that

The difference of the logarithms of two consecutive numbers is less in proportion as the numbers themselves are greater.

203. When we have calculated a table of logarithms for any base a, we can easily change the system, and calculate another table for a new base b.

Let cb', a is the log. of c in the system whose base is };
Taking the logs. in the known system, whose base is a, we have

The log. of c in the system whose base is b, is the quotient arising from dividing the log. of c by the log. of the new base b, both these last logs. being taken in the system whose base is a.

In order

1

to have x the log. of c in the new system, we must multiply log. c by is constant for all numbers, and is log. b; this last factor log. b called the Modulus; that is to say, if we divide the logs. of the same number c taken in two systems, the quotient will be invariable for these systems, whatever may be the value of c, and will be the modulus, the constant multiplier which reduces the first system of logs. to the second.'

If we find it inconvenient to make use of a log. calculated to the base 10, we can in this manner, by aid of a set of tables calculated to the base 10, discover the logarithm of the given number in any required system.