§2. Series expansions
349
for x E ] - 1,1[, and, since this power series is itself the derivative of
x - x 2 /2 + x 3 /3 - ... , one concludes, as in Chap. III, nO 16, example 1, that
(10.6)
log(1 + x) = x - x 2 /2 + x 3 /3 - ... for x E ]- 1,1]
(the case x = 1 is obtained by passing to the limit).
In its turn formula (5) implies a result which we shall extend in the next
nO to the case where x is complex:
(10.7)
exp(x) = lim(1 + x/n)n
for all x E JR and even
(10.7')
Indeed
exp(x) = lim(1 + hX)l/h when h ~ o.
log [(1 + hX)l/h] = log(1 + hx)/h = x log (1 + ~:) -logl,
and since hx tends to 0, the fraction in the third term tends to the derivative
of the function log at x = 1, i.e. to 1, by (5). In consequence, log [(1 + hX)l/h]
tends to x, whence, by continuity,
exp(x) = limexp {log [(1 + hX)l/h]} = lim(1 + hX)l/h,
which proves (7').
To conclude, let us remark that the functions log and exp can be used to
define the general exponentials aX and to establish their properties. The addition formula for the function log shows immediately that log( an) = n.log a
for nEZ, i.e. that
an = exp(n.loga).
One deduces that
exp [log(a)p/q]q = exp(p.log a) = a P ,
whence
(a P )l/q = exp[log(a)p/q].
From this it is natural to define
(10.8)
aX = exp(x.loga)
for all x E JR, or by
(10.9)
log(a X ) = x.loga.