and therefore series (1) is equal to series (2) however great n may be. Hence if n be indefinitely increased, we have

The series in the parenthesis is usually denoted by e;

which is the expansion of e* in powers of x.
This result is called the Exponential Theorem.
If we put x= 1, we have from (3)

(3)

From this series we may readily compute the approximate value of e to any required degree of accuracy. This constant value e is called the Napierian base (Art. 64). To ten places of decimals it is found to be 2.7182818284.

Cor. Let ae; then c= log, a, and a = ecx. Substituting in (3), we have

which is the expansion of a in powers of x.

130. Logarithmic Series. To expand log.(1+x) in a series of ascending powers of x.

=1+x[α-1 − 1 ( a − 1)2 + } (a−1)3- (a−1)*+ ···] −1 · }

+terms involving a2, a3, etc.

Comparing this value of a with that given in (4) of Art. 129, and equating the coefficients of x, we have

log, a = a − 1 − 1 ( a − 1)2 + } ( a − 1)3 — — (a − 1)1 + ···

Put a = 1+x; then

This is the Logarithmic Series; but unless a be very small, the terms diminish so slowly that a large number of them will have to be taken; and hence the series is of little practical use for numerical calculation. If x> 1, the series is altogether unsuitable. We shall therefore deduce some

and (3) becomes

n+1 loge

n

=

or log, (n + 1)

n+1

1

1

2n+1'

1

= 2 (2 m2 + 1 + 3 (2m2 + 1)2 + 5 (2 m2 + 1)2 + ],

2n+1 3(2n+1)35(2n+1)3

1

1

=log, n+2

1

+

2n+1*3(2n+1)3*5(2n+1)5'

This series is rapidly convergent, and gives the logarithm of either of two consecutive numbers to any extent when the logarithm of the other number is known.

131. Computation of Logarithms. - Logarithms to the base e are called Napierian Logarithms (Art. 64). They are also called natural logarithms, because they are the first logarithms which occur in the investigation of a method of calculating logarithms. Logarithms to the base 10 are called common logarithms. When logarithms are used in theoretical investigations, the base e is always understood, just as in all practical calculations the base 10 is invariably employed. It is only necessary to compute the logarithms of prime numbers from the series, since the logarithm of a composite number may be obtained by adding together the logarithms of its component factors. The logarithm of 10. Putting n = 1, 2, 4, 6, etc., successively, in (4) of Art. 130, we obtain the following

1

1

1

log, 3 log, 2+2 =+ + + +

3.53 5.55 7.57

...

=

+ + +
9 3.93 5.95 7.97

= 1.94590996.

13

3.133

5.135

The number of terms of the series which it is necessary to include diminishes as n increases. Thus, in computing

the logarithm of 101, the first term of the series gives the result true to seven decimal places.

By changing 6 to 10 and a to e in (1) of Art. 65, we have

or

common log m = Napierian log m × .43429448.

The number .43429448 is called the modulus of the common system. It is usually denoted by μ.

Hence, the common logarithm of any number is equal to the Napierian logarithm of the same number multiplied by the modulus of the common system, .43429448.

Multiplying (4) of Art. 130 by μ, we obtain a series by which common logarithms may be computed; thus,

log 10 2

Common Logarithms.

= μ log, 2 = .43429448 × .69314718.3010300. log 10 3 =μlog, 3.43429448 × 1.09861228.4771213. log1042 log102

e

= .6020600.

log105 μ log, 5 = .43429448 × 1.60943790 = .6989700. And so on.

132. If be the Circular Measure of an Acute Angle, sin 0, 0, and tan ✪ are in Ascending Order of Magnitude.

With centre O, and any radius, describe an arc BAB'. Bisect the angle BOB' by OA; join BB', and draw the tangents BT, B'T.

Let AOBAOB' = 0. Then

BB'arc BAB' < BT + B'T

(Geom., Art. 246)

B

B

T

A

.. BC <arc BA <BT.

Now as is diminished indefinitely, sec limit unity; then when = 0, we have sec

approaches the

=1.

NOTE. From this it follows that the sines and the tangents of very small angles are proportional to the angles themselves.

134. If 0 is the Circular Measure of an Acute Angle, sin