BINOMIAL EXPANSIONS.

143

Thus it appears (10) that the sum of the coefficients of any binomial power is equal to 2 raised to the same power, and (2°) that the sum of the coefficients of any binomial power taken alternately plus and minus is equal to zero. This furnishes a convenient method of verifying an involution. For example,

(a+x)1=1•a+1.x, 1+1=2=21, and +1-1=0; (a+x)2= a2+2ax + x2, 1+2+1 = 4 = 22, and +1−2+1=0; (a + x)3 = a3 +3a2x+3ax2+x3, 1+3+3+1= 8 = 23, and 1-3+3-1=0.

Expand (a+x)*, (a + x)3, (a + x), (a + x)', by aid of (250), and (a-x)', (a-x), (a — x), (a−x)', by aid of (251), and verify.

If in (250) and (251), we make n =

1

there results

from which, substituting z2 for x, we obtain

(c2 +22)*=c+

3.5z8

(257)

(258)

24.2.3.4c7

changing into in (258),

(c2 — x)‡ :

=c

The student may operate like changes upon (256), and investigate analogous forms for higher roots, as the 4th, 5th, &c. These expansions may be employed for the extraction of the roots of numbers. Thus, let the square root of 101 be required. We have (258)

Find √102, 103, √104; √99, √98, √97, 3/1001, &c.

We may put (250) under a new form, for dividing by a", there

LOGARITHMS.

LOGARITHMS.

145

Definition. Let a3 = x,

(263)

then is y denominated the Logarithm of x. The constant a is called the base of the system.

PROPOSITION II.

The Logarithm of a product, consisting of several fac- (264) tors, is equal to the sum of the Logarithms of those factors.

For, let x = X1, X2, X3, ...; Y = Y19 Y29 Y39

then (263)

or* _ _L(x1 • X2 • X3 ...) = (Y1+Y2+Y3 + ...) = Lx1 + Lx2

+ LX3+....

Q. E. D.

Cor. 1. The logarithm of the nth power of any number (265) is equal to n times the logarithm of the number itself.

Cor. 2. The logarithm of the nth root of any number, (266) is equal to the nth part of the logarithm of the number itself.

Cor. 3. The logarithm of a fraction is equal to the loga- (267) rithm of the numerator diminished by the logarithm of the denominator.

Scholium. We perceive that addition of logarithms corresponds to multiplication of numbers, subtraction to division, multiplication to involution, and division to evolution. We have, then, only to possess a "Table of Logarithms," calculated to a given base,

* L, logarithm of.

=

say a 10, in order to perform numerical operation with remarkable facility. Thus, to obtain the cube root of 2, nothing more would be necessary than to take the logarithm of 2, divide by 3, and seek the corresponding number from the table.

whereby we are enabled to solve, with the utmost facility, a numerical equation in which the exponent is the unknown quantity.

PROPOSITION III.

A number being given, it is required to find a form by which we may calculate its logarithm.

Since the relation a3 = x, gives y a function of x, let us endeavor to expand y in terms of x. To this end we proceed to determine the derivative of y.

Changing x into x + h and y into y+k, we have

in which, substituting x for a" and 1+b for a, in order to subject a to the influence of the Binomial Theorem, we have (262)

h = x[(1+b)* − 1] = x [1+kb+k(k − 1) •

2

dividing k by each member of this equation, observing that 1,-1, cancel each other, and that k then becomes a factor common to the numerator and denominator of the second fraction, there results

from which, observing k = 0 when h becomes

=

0, we have

Applying the rule of (246), in order to return from the derivative (268) to the primitive function, we fall upon the equation

from which a disappears, and nothing can be inferred-save that the logarithm of a number cannot be developed in terms of the number simply.

But if we make y the logarithm of 1 + x instead of x, and repeat the above operation, from

or (262), y' = M [1+(−1)x + (− 1) (− 1 − 1) — 22 + (− 1) (− 1 − 1)

from which, returning to the function, we get

y = M[x − x2 +‡x3 — ‡x1 +‡x3 − ‡x® +, −, ...] + constant.

In order to determine the constant, we observe that ≈ and y vanish together, since y = 0 in a = 1+x, gives 1+ x = ao = 1, and .. x = 0; therefore, substituting these corresponding values of y and x, we get

0= M • 0+ constant, .. constant = 0, and L(1+x)=y=M[x − ‡x2 +‡x3 — ‡x1 + £x3 −‡x® +,−,...], (270)

a logarithmic series, in which the logarithm of any number 1+x is expressed in terms of a number less by unity, x.

The constant, M, depending upon the base, a, (269), is denominated the Modulus of the system. Taking a different base, a, we obtain a new modulus,