Second, To divide quantities by means of their logarithms.

Find the logarithm of the dividend and the logarithm of the divisor, from a table; subtract the latter from the former, and look for the number corresponding to this difference; this will be the quotient required.

Third, To raise a number to any power.

Find from a table the logarithm of the number, and multiply it by the exponent of the required power; find the number corresponding to this product, and it will be the required power.

Fourth, To extract any root of a number.

Find from a table the logarithm of the number, and divide this by the index of the root; find the number corresponding to this quotient, and it will be the root required.

By the aid of these principles, we may write the following equivalent expressions :

Log (am.b.c....) = m log a + n log b + p log c+....

234. We have already explained the method of determining the characteristic of the logarithm of a decimal fraction, in the common system, and by the aid of the principle demonstrated in Art. 231, we can show

That the decimal part of the logarithm is the same as the decimal part of the logarithm of the numerator, regarded as a whole number. For, let a denote the numerator of the decimal fraction, and let m denote the number of decimal places in the fraction, then will the fraction be equal to

and its logarithm may be

α

α

10m3

expressed as follows:

=

log 10m = log a - log (10) log am log 10 = log am, but m is a whole number, hence the decimal part of the loga rithm of the given fraction is equal to the decimal part of log a, or of the logarithm of the numerator of the given fraction.

Hence, to find the logarithm of a decimal fraction from the common table,

Look for the logarithm of the number, neglecting the decimal point, and then prefix to the decimal part found a negative charac teristic equal to 1 more than the number of zeros which immediately follow the decimal point in the given decimal.

other

The rules given for finding the characteristic of the logarithms taken in the common system, will not apply in system, nor could we find the logarithm of decimal fractions

any

directly from the tables in any other system than that whose base

is 10.

These are some of the advantages which the common system possesses over every other system.

235. Let us again resume the equation

a = N.

1st. If we make N=1, x must be equal to 0, since ao=1; that is,

The logarithm of 1 in any system is 0.

2d. If we make Na, x must be equal to 1, since a1 = that is,

Whatever be the base of a system, its logarithm, taken in that system, is equal to 1.

Then, when N = 1, x = 0; when N> 1, x>0; when N<1, <0, or negative; that is,

In any system whose base is greater than 1, the logarithms of all numbers greater than 1 are positive, those of all numbers less than 1 are negative.

If we consider the case in which N<1, we shall have

Now, if N diminishes, the corresponding values of a must increase, and when N becomes less than any assignable quantity, or 0, the value of x must be : that is,

The logarithm of 0, in a system whose base is greater than 1, is equal to ∞o.

Second, suppose a <1.

Then, when N=1, z=0; when N<1, >0; when N>1, <0, or negative: that is,

In any system whose base is less than 1, the logarithms of all numbers greater than 1 are negative, and those of all numbers less than 1 are positive.

If we consider the case in which N< 1, we shall have a2 = N, in which, if N be diminished, the value of x must be increased; and finally, when N 0, we shall have x = ∞: that is,

=

The logarithm of 0, in a system whose base is less than 1, is equal to +∞.

Finally, whatever values we give to x, the value of a or N will always be positive; whence we conclude that negative numbers have no logarithms.

Logarithmic Series.

236. The method of resolving the equation,

ax

b,

explained in Art. 226, gives an idea of the construction of logarithmic tables; but this method is laborious when it is necessary to approximate very near the value of x. Analysts have discovered much more expeditious methods for constructing new tables, or for verifying those already calculated. These methods consist in the development of logarithms into series.

If we take the equation,

axy,

and regard a as the base of a system of logarithms, we shall have,

log y = x.

The logarithm of y will depend upon the value of y, and also upon a, the base of the system in which the logarithms are taken.

Let it be required to develop log y into a series arranged according to the ascending powers of y, with co-efficients that are independent of y and dependent upon a, the base of the system.

Let us first assume a development of the required form,

log y = M + Ny + Py2 + Qy3 + &c.,

in which M, N, P, &c. are independent of y, and dependent upon a. It is now required to find such values for these coefficients as will make the development true for every value

of y.

Now, if we make y = 0, log y becomes infinite, and is either negative or positive, according as the base a is greater or less than 1, (Arts. 234 and 235). But the second member under this supposition, reduces to M, a finite number: hence, the development cannot be made under that form.

Again, assume,

logy My+ Ny2+ Py3 + &c.

=

If we make y = 0, we have

log 0 = 0 that is, ∞ = 0,

which is absurd, and therefore the development cannot be made under the last form. Hence, we conclude that,

The logarithm of a number cannot be developed according to the ascending powers of that number.

Let us write (1+ y), for y in the first member of the assumed development; we shall have,

log (1+ y) = My + Ny2 + Py3 + Qy1+ &c.

(1),

making y0, the equation is reduced to log 10, which does. not present any absurdity.

Since equation (1) is true for any value of y, we may write z for y; whence,

(2).

log (1 + 2) = Mz + Nz2 + P23 + Q2a + &c. Subtracting equation (2) from equation (1), member from mem

ber, we obtain,

log (1 + y) — log (1 + z) = M(y − 2) + N(y2 — z2) ·† P(y3 — 23)