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.. 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),

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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

Nn = anx

of n.

.. 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 nth root of both members of equation (1).

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X
n

.. 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,

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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:

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1o. If a⇒ 1, making x = 0, we have N1; the hypothesis x = 1 gives Na. 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 Na-, or N =

Here, as x 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 b'

2o. If a 1, put a = where b⇒1, and we shall then have N =

or N = b2, 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 a 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 be 1, the logarithms of numbers > 1 are positive, and the logarithms of numbers ≤ 1 are negative. The contrary takes place if the base

be 41.

III. The base being fixed, any number has only one real logarithm; but the same 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 92 = 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- ∞ up to +∞, we find that the corresponding values of N are positive numbers only, from 0 up to +∞.

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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,

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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 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 = ax

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

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If we have an equation a2 = b, where z depends upon an unknown quantity, and we have

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K some known number, the problem depends upon the

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an equation of the second degree, from which we find x = 2, x = 3.

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(m log. c + log. ƒ) x2 (n log. b + p log. f)x + a log. b

a quadratic equation, from which the value of x may be determined.

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= 0

Equations of this nature are called Exponential Equations.

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

N

log. (N+1) — log. N = log. (+1) = log. (1 + 1/7)

a quantity which approaches to the logarithm of 1, or zero, in proportion as decreases, that is, as N increases. Hence it appears, that

1

N

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, x is the log. of c in the system whose base is 3;

Taking the logs. in the known system, whose base is a, we have

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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

log. c by

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log. b

to have x the log. of c in the new system, we must multiply this last factor is constant for all numbers, and is log. 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.

For example, let it be required, by aid of Briggs' tables, to find the log. of

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Taking these logs. in Briggs' system, and reducing, we find

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which is manifestly the true result; for in this case the general equation

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for, by the definition of a log. in the equation na3, x is the log. n.

In like manner,

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(1.) x=1.584962, or x=log. (—4)÷log. 2.

(2.) x={a+log. n÷log. m} and y={a-log. n÷log. m}.

(3.) x=log. a÷(log. m+log. n) and y=log. a÷(log. m+log, n).

h

k

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