Hence, if all the factors of a given number, in any case of this kind, be supposed equal to each other, and the sum of them be denoted by m, the preceding property will then bo、 me

log. ym=m log. y.

From which it appears, that the logarithm of the mth power of any number is equal to m times the logarithm of that number.

In like manner, if the equation a*=y be divided by a*= y', we shall have, from the nature of powers, as before, =2; and by the definition of logarithms,

at

or axx'=

y

y

laid down, in the first part of this article, x-x'= log.

Hence the logarithm of a fraction, or of the quotient arising from dividing one number by another, is equal to the logarithm of the numerator minus the logarithm of the denominator.

And if each member of the common equation ax=y be raised to the fractional power denoted by - we shall

m

n

And, consequently, by taking the logarithms, as before, m -x= log. yn, or log yn log.y

m

m

1 n

Where it appears, that the logarithm of a mixed root, or power, of any number, is found by multiplying the logarithm of the given number by the numerator of the index of that power, and dividing the result by the denominator.

And if the numerator m, of the fractional index, be,.

· 203

in this case, taken equal to 1, the above formula will then become

From which it follows, that the logarithm of the nth root of any number is equal to the nth part of the logarithm of that number.

Hence, besides the use of logarithms in abridging the operations of multiplication and division, they are equally applicable to the raising of powers and extracting of roots; which are performed by simply multiplying the given logarithm by the index of the power, or dividing it by the number denoting the root.

But, although the properties here mentioned are common to every system of logarithms, it was necessary, for practical purposes, to select some one of them from the rest, and to adapt the logarithms of all the natural numbers to that particular scale.

And as 10 is the base of our present system of arithmetic, the same number has accordingly been chosen for the base of the logarithmic system, now generally used.

So that, according to this scale, which is that of the common logarithmic tables, the numbers

104, 10-3, 102, 101, 10°, 101, 102, 103, 104, &c.

1, 10, 100, 1000, 10000, &c.

10000' 1000' 100' 10'

have for their logarithms

-4, -3, -2, -1, 0, 1, 2, 3, 4, &c.

Which are evidently a set of numbers in arithmetical progression, answering to another set in geometrical progression; as is the case in every system of logarithms.

And therefore, since the common, or tabular, logarithm of any number (n) is the index of that power of

10, which when involved, is equal to the given number it is plaiu, from the following equation,

that the logarithms of all the intermediate numbers, in the above series, may be assigned by approximation, and made to occupy their proper places in the general scale.

It is also evident, that the logarithms of 1, 10, 100, 1000, &c. being 0, 1, 2, 3, &c. respectively, the logarithm of any number, falling between 0 and 1, will be 0 and some decimal parts; that of a number between 10 and 100, 1 and some decimal parts; of a number between 100 and 1000, 2 and some decimal parts; and so on, for other numbers of this kind.

1

And for a similar reason, the logarithms of

1 1

10' 100'

&c. or of their equals .1, .01, .001, &c. in the 1000 descending part of the scale, being -1, -2, -3, &c. the logarithm of any number, falling between 0 and 1, will be -1, and some positive decimal parts; that of a number between.1 and .01, 2 and some positive decimal parts; of a number between .01 and .001,-3, and some positive decimal parts; &c.

Hence, likewise, as the multiplying or dividing of any number by 10, 100, 1000, &c. is performed by barely increasing or diminishing the integral part of its logarithm by 1, 2, 3, &c. it is obvious that all numbers, which consist of the same figures, whether they be integral, fractional, or mixed, will have, for the decimal part of their logarithms, the same positive quantity.

So that, in this system, the integral part of any logarithm, which is usually called its index, or characteristic, is always less by 1 than the number of integers which the natural number consists of; and for decimals,

it is the number which denotes the distance of the first significant figure from the place of units.

Thus, according to the logarithmic tables in common use, we have

Where the sign

2.7891575

&c.

is put over the index, instead of before it, when that part of the logarithm is negative, in order to distinguish it from the decimal part, which is always to be considered as +, or affirmative.

Also, agreeably to what has been before observed, the logarithm of 38540 being 4.5859117, the logarithms of any other numbers, consisting of the same figures, will be as follows:

Which logarithms, in this case, as well as in all others of a similar kind, whether the number contains ciphers or not, differ only in their indices, the decimal, or positive part, being the same in them all. (e)

(e) The great advantages attending the common, or Briggean system of logarithms, above all others, arise chiefly from the readi

T

And as the indices, or integral parts, of the logarithms of any numbers whatever, in this system, can always be thus readily found from the simple consideration of the rule above mentioned, they are generally omitted in the tables, being left to be supplied by the operator, as occasion requires.

It may here, also, be farther added, that, when the logarithm of a given number, in any particular system, is known, it will be easy to find the logarithm of the same number in any other system, by means of the following equations,

ax==n, and ex'=n; or log. n=x, and l n=x'.

Where log. denotes the logarithm of n, in the system of which a is the base, and 1. its logarithun in the system of which e is the base.

For, since axex', or a=e, and e* =a, we shall have,

and for the base e,1. a, or x'=x l. a.

x

Whence, by substitution, from the former equations,

log, n=1. nxlog. e; or log. n=l. nX:

1

1.a

Where the multiplier log. e, or its equal

ness with which we can always find the characteristic or integral part of any logarithm from the bare inspection of the natural number to which it belongs, and the circumstance, that multiplying or dividing any number by 10, 100, 1000, &c only influences the characteristic of its logarithm, without affecting the decimal part. Thus, for instance, if i be made to denote the index, or integral part of the logarithm of any number N, and d its decimal part, we shall have log Ni+d; log. 10m× N~(i+m)+d; log.

N

10m

(im)+d; where it is plain that the decimal part of the logarithm, in each of these cases, remains the same.