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

log. y* = log, y


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 . . 10-4, 10-3, 10-2, 10-1, 10", 10", 102, 103, 104, &c.

Or, 1

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 plain, from the following equation,

1 10%

=n, or 10-2 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 1 1 And, for a similar reason, the logarithms of

10' 100% 1000 &c., or of their equals .1, .01, .001, &c., in the 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

Numbers. | Logarithms.
1.368201 0.1361496
20.0500 1.3021144
335.260 | 2.5253817
.465:21 1.6676490
.06154 | 2.7891575


Where the sign is put over the index, instead of before it, then that part of the logarithm is negative, in order to distinguish it from the decimal part, which is always to be considered as t, 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:

Numbers. Logarithms.

3854 3.5859117 385.4 | 2.5859117 38.54 | 1.5859117 3.854 1 0.58591119 .3854 1.5859117 .03854 | 2.5859117

.003854 | 3.5859117 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.*

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 abovementioned, 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 :

au n, and el! = n, or log. n = x, and l. n= de'. Where log. denotes the logarithm of n, in the system of which a is the base, and l. its logarithm in the system of which e is the base.


For, since aim ex', or all! = e, and e = a, we shall have for the base a, - log. e, or x = alog. e ;


ac' and for the base e,

Elia, or ac'

a l. a.

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* The great advantages attending the common, or Briggean system of logarithms, above all others, arise chiefly from the readiness 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, n=i+d; log:. 10m. X N= (i-+-m) +d; log: :(im)+d; where it is plain that the decimal part of

10m the logarithm, in each of these cases, remains the same.



Whence, by substitution from the former equations

log. n-=l.n X log. e; or log. n =
n l.n X


1 Where the multiplier, log. e, or its equal

expresses the constant relation which the logarithms of n have to each other in the systems to which they belong.

But the only system of these numbers deserving of notice, except that above described, is the one that furnishes what have been usually called hyperbolic or Naperian logarithms, the base e of which is 2.718281828459

Hence, in comparing these with the common or tabular logarithms, we shall have, by putting a in the latter of the above formulæ 10, the expression log.n=l.nx or l. n log. n x 1.10.

1.10' Where log. in this case denotes the common tabular logarithm of the number n, and l its hyperbolic logarithm; the con

1 stant factor, or multiplies, which is


2,3025850929 equal .4342944819, being what is usually called the modulus of the common system of logarithms.*

PROBLEM 1.-To compute the logarithm of any of the natural numbers 1, 2, 3, 4, 5, &c.

RULE 1.--1. Take the geometrical series 1, 10, 100, 1000, 10000, &c., and apply to it the arithmetical series, 0, 1, 2, 3, 4, &c., as logarithms.

2. Find a geometric mean between 1 and 10, 10 and 100, or any other two adjacent terms of the series, betwixt which the number proposed lies.

3. Also, between the mean, thus found, and the nearest extreme, find another geometrical mean in the same manner;


or its

* It may here be remarked, that although the common logarithms have superseded the use of hyperbolic or Naperian logarithms, in all the ordinary operations to which these numbers are generally applied, yet the latter are not without some advantages peculiar to themselves; being of frequent occurrence in the application of the Fluxionary Calciuus, to many analytical and physical problems, where they are required for the finding of certain fluents, which could not be so readily determined without their assistance; on which account great pains have been taken to calculate tables of hyperbolic logarithms, to a considerable extent, chiefly for this purpose. Mr. Barlow, in a Collection of Mathematical Tables lately published, has given them for the first 10000 numbers.

and so on, till you are arrived within the proposed limit of the number whose logarithm is sought.

4. Find, likewise, as many arithmetical means between the corresponding terms of the other series 0, 1, 2, 3, 4, &c., in the same order as you found the geometrical ones, and the last of these will be the logarithm answering to the number required.



Let it be required to find the logarithm of 9.
Here the proposed number lies between 1 and 10.
First, then, the log. of 10 is 1, and the log. of 1 is 0.

There (10 X 1)=V 10 = 3.1622777 is the geometrical mean;

And I (1+0) ==.5 is the arithmetical mean;
Hence the log. of 3.1622777 is .5.
Secondly, the log. of 10 is 1, and the log. of 3.1622777 is .5.

Therefore ✓ (10 X 3.1622777) = 5.6234132 is the geometrical mean;

And I (1 +.5)=.75 is the arithmetical mean;
Hence the log. of 5.6234132 is .75.
Thirdly, the log, of 10 is 1, and the log, of 5.6234132 is

Therefore V (10 X 5.6234132) = 7.4989422 is the geometrical mean;

And I (1 +.75) =-.875 is the arithmetical mean;
Hence the log. of 7.4989422 is .875.

Fourthly, the log. of 10 is 1, and the log. of 7.498422 is 875;

Therefore ✓(10 X 7.4989422) = 8.6596431 is the geometrical mean;

And I (1 +.875) = .9375 is the arithmetical mean;
Hence the log. of 8.6596431 is .9375.

Fifthly, the log. of 10 is 1, and the log. of 8.6596431 is .9375.

Therefore ✓(10 X 8.6596431)= 9.3057204 is the geometrical mean.

And I (1 + :9375) = .96875 is the arithmetical mean;
Hence the log. of 9.3057204 is .96875.

Sixthly, the log. of 8.6596431 is .9375, and the log. of 9.3057204 is .96875;

Therefore v (8.6596431 X 9.3057204) = 8.9768713 is the geometrical mean.

"And 1 (.9375 +.96875) = .953125 is the arithmetical

mean ;

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