Hence, for all positive values of x, {ƒ (1)}* = ƒ (x).

Lastly, let a be negative, and equal to -y, so that y is positive; then ƒ (− y) ׃ (y)=ƒ(0) from (i); but ƒ (0) = 1, therefore ƒ (− y) = 1/ ƒ (y).

Hence, equating the coefficients of " in the expansions of the two expressions for (e* - 1)", we have

Hence, equating coefficients of " in the two expressions for (ex- ebx)", we have

If we put na=x and b-a=y, the last result becomes

We have also, if k be any positive integer less than n,

The following particular cases are of importance, k being less than n.

3k to n+1 terms=0,

Ex. 2. Shew that the limit when n is infinite of (1+

Ex. 3. Shew that

of (1+) is c

α

Ex. 12. Shew that the coefficient of x" in the expansion of

302. Definition. The index of the power to which one number must be raised to produce a second number is called the logarithm of the second number with respect to the first as base. Thus, if a* =y, then x is called the logarithm of y to the base a, and this is expressed by the notation x = loga y.

We proceed to investigate the fundamental properties of logarithms, and to shew how logarithms can be found, and how they can be employed to shorten certain approximate calculations.

303. Properties of Logarithms. The following are the fundamental properties of logarithms.

I. Since a = 1, for all values of a, it follows that log, 1=0.

be.

Thus the logarithm of 1 is 0, whatever the base may

Thus the logarithm of a product is the sum of the logarithms of its factors.

then

III. If

log1 = a, and log。 y = ß ;

x = aa, y=a3, and.. x÷y=aa-ẞ;

.. loga (x÷y)= a - ẞ=log, x— loga y.

Thus the logarithm of a quotient is the algebraic difference of the logarithms of the dividend and the divisor.

IV. If x = aa; then xm =ama, for all values of m.

Hence log, x = ma = m log. x.

Thus the logarithm of any power of a number is the product of the logarithm of that number by the index of the

power.

V. Let log, x=a, and log, x=ẞ; then x=aa = bß ;

Hence the logarithm of any number to the base b will be found by multiplying the logarithm of that number to the base a by the constant multiplier log, a.

304. The logarithmic series. Let ae, so that k=log, a; then a = ex = ex loge a.

we have

a = ex loge a = 1+x log, a +

Hence from Art. 298,

{x log. (1+y)}2+.

+...

Now, provided y be numerically less than unity, (1+ y)* can be expanded by the binomial theorem; we then have

1

2

=1+x log, (1+ y) + {x log, (1 + y)}2 +.....

y'+...

The series on the right is convergent for all values of x and y, and the series on the left is convergent for all values of a provided y is numerically less than unity. Hence, for such values of y, we may equate the coefficients of a on the two sides of the equation. We thus obtain