346
IV - Powers, Exponentials, Logarithms, Trigonometric Functions
(9.9)
y = log x ¢=:::> x = e Y ¢=:::> x = exp y = L yn In!,
clearly a fundamental result, as is the relation
(9.10)
log' x = 1/x
that one obtains on putting a = e in (8.2), and already proved otherwise in
Chap. II, nO 10.
We have also proved (9) in Chap. III, n° 2 and (10) at the end of nO 14,
using the integral
10gb -loga = 1b dtlt
of Chap. II, nO 11. Starting from here and from the formula exp' = exp, one
sees that the derivative of the function exp(1ogx), or of f(x) = log(expx),
is equal to 1. We must then have f(x) = x up to an additive constant, by
Chap. III, n° 16, Corollary 1 of the mean value theorem, whence f(x) = x
since f(O) = O.
You will find another method in n° 10.
The functions log and exp thus satisfy all the identities obtained in nO 4;
in particular,
(9.11)
for all x > 0 and all s E R Since, on the other hand, the functions loga are
all mutually proportional, they are proportional to the N apierian log:
(9.12)
loga(x) = log(x)/log(a)
since the left hand side is 1 for x = a.
Further, all the exponential functions can be expressed in terms of the
single function exp: if we replace x by a and s by x in (11), we have
(9.13)
eXPa(x) = exp(x.loga).
In other words, there are no exponential functions apart from the functions
x f--+ e CX where c is an arbitrary real constant, and no logarithmic functions
other than the functions x f--+ c.log x. From now on you may forget loga
and the other eXPa without any worry: they will never appear, except, with
a == 10, in numerical calculations.
10 - Exponential and logarithmic series: direct method
To obtain the main result of the preceding nO - the fact that the functions
log and exp are inverses of one another -, we drew on the direct construction
of the expressions aX and on the theorems characterising the logarithmic and
exponential functions by their functional equations.
IV - Powers, Exponentials, Logarithms, Trigonometric Functions
(9.9)
y = log x ¢=:::> x = e Y ¢=:::> x = exp y = L yn In!,
clearly a fundamental result, as is the relation
(9.10)
log' x = 1/x
that one obtains on putting a = e in (8.2), and already proved otherwise in
Chap. II, nO 10.
We have also proved (9) in Chap. III, n° 2 and (10) at the end of nO 14,
using the integral
10gb -loga = 1b dtlt
of Chap. II, nO 11. Starting from here and from the formula exp' = exp, one
sees that the derivative of the function exp(1ogx), or of f(x) = log(expx),
is equal to 1. We must then have f(x) = x up to an additive constant, by
Chap. III, n° 16, Corollary 1 of the mean value theorem, whence f(x) = x
since f(O) = O.
You will find another method in n° 10.
The functions log and exp thus satisfy all the identities obtained in nO 4;
in particular,
(9.11)
for all x > 0 and all s E R Since, on the other hand, the functions loga are
all mutually proportional, they are proportional to the N apierian log:
(9.12)
loga(x) = log(x)/log(a)
since the left hand side is 1 for x = a.
Further, all the exponential functions can be expressed in terms of the
single function exp: if we replace x by a and s by x in (11), we have
(9.13)
eXPa(x) = exp(x.loga).
In other words, there are no exponential functions apart from the functions
x f--+ e CX where c is an arbitrary real constant, and no logarithmic functions
other than the functions x f--+ c.log x. From now on you may forget loga
and the other eXPa without any worry: they will never appear, except, with
a == 10, in numerical calculations.
10 - Exponential and logarithmic series: direct method
To obtain the main result of the preceding nO - the fact that the functions
log and exp are inverses of one another -, we drew on the direct construction
of the expressions aX and on the theorems characterising the logarithmic and
exponential functions by their functional equations.