An introduction to the study of the elements of the differential and integral calculus. From the German of the late Axel Harnack, With the permission of the author.

~ 7. 8.

The logarithlm.

13

Multiplying 1 + q continually on itself to M times produces a number
which is certainly greater than 1 + Mil; therefore
A > 1 + Mq, < < M ---1
But if A < 1, let us put
-— A -=, A- — 1- — 1 + -;
then
1
A < - therefore d <
Now if we consider, that by the procedure just explained the series
A1, AE2,... A%,...
also can be replaced by the series
a a A'2I, o 11
when Lim (a,,) = A, the result of this investigation can be written
in the form:
(VI) (Lim aCi) Lim (9,,) == Lim (a,89).
8. Inversion of involution presents a second problem: that of
the logarithm. Two positive numbers A and B are given, each
defined as limiting value of a series; it is required to determine a
number x having the property that B = A. Here x is called the
logarithm (exponent) of the number A (number) with regard to the
base B, and is written
x = B log A.
It can be shown, in the first place that only one number x has
this property - for we cannot at the same time have (B < 1)
Bx
B - A - Bx' and so BX' — 1
Bx
without having x - x'= 0, - and moreover in the second place that
this number x can be expressed as limiting value of a series. In fact,
forming the series of terms
~ B-2 B-, i, B', B2,.
there will be among them two values, such that, if B > 1
Bi < A < BW+1.
If we interpolate rational fractions between A and A - + we have:
2+2'
B + < A < B
As we increase the values of the denominator we obtain two series
by which x is defined.