§ 1. Direct construction
341
So we see that the ratio (1) truly has a limit, say c, as h tends to 0 through
values> O. Now consider what happens when h tends to 0 through negative
values. We can put h = -k with k > 0, whence
l/ak-l
_k ak - l
-k
=a -k-;
when h tends to 0 through negative values, the factor a- k tends to 1, and
the last fraction to c by the first case. So we find a left limit equal to the
right limit, which completes the proof of the existence of the derivative at
the origin, and so everywhere, of the function aX.
These calculations have demonstrated that there is a formula
(7.5)
with a factor L(a) independent of x, but of course depending on a. This is
the derivative at the origin, so that
(7.6)
L(a) = lim (a h - l)lh.
h--O,h,cO
Since (ab)X = aX b X , the product rule for differentiation shows that
L(ab)(ab)X = L(a)aXb X + aX L(b)b X ,
whence L(ab) = L(a) + L(b), curiously. Moreover, one can take h = lin and
let the integer n tend to infinity; one finds
(7.7)
L(a) = limn (a 1 / n -1) for all a > O.
In other words, L(x) = log x and in consequence
(7.8)
Even though we carefully banished the Napierian log in the preceding nOs, we
cannot prevent it from returning at the gallop; this shows the privileged status
of the so-called natural log relative to the log to any other base. Further,
formula (6) is much more general than the definition (7) of the log since
one can now substitute for lin anything tending to 0 in a "discrete" or
"continuous" way. In the author's youth (he no longer enters competitions
nor judges them, nor does he know what happens nowadays)8 this remark
gave rise to abundantly exploited traps. Consider for example the sequence
8 The university examinations suffice. Peano once said that "relations between
students and professors would be excellent if it were not for the examinations,
which force students to consider their masters as potential judges" .
341
So we see that the ratio (1) truly has a limit, say c, as h tends to 0 through
values> O. Now consider what happens when h tends to 0 through negative
values. We can put h = -k with k > 0, whence
l/ak-l
_k ak - l
-k
=a -k-;
when h tends to 0 through negative values, the factor a- k tends to 1, and
the last fraction to c by the first case. So we find a left limit equal to the
right limit, which completes the proof of the existence of the derivative at
the origin, and so everywhere, of the function aX.
These calculations have demonstrated that there is a formula
(7.5)
with a factor L(a) independent of x, but of course depending on a. This is
the derivative at the origin, so that
(7.6)
L(a) = lim (a h - l)lh.
h--O,h,cO
Since (ab)X = aX b X , the product rule for differentiation shows that
L(ab)(ab)X = L(a)aXb X + aX L(b)b X ,
whence L(ab) = L(a) + L(b), curiously. Moreover, one can take h = lin and
let the integer n tend to infinity; one finds
(7.7)
L(a) = limn (a 1 / n -1) for all a > O.
In other words, L(x) = log x and in consequence
(7.8)
Even though we carefully banished the Napierian log in the preceding nOs, we
cannot prevent it from returning at the gallop; this shows the privileged status
of the so-called natural log relative to the log to any other base. Further,
formula (6) is much more general than the definition (7) of the log since
one can now substitute for lin anything tending to 0 in a "discrete" or
"continuous" way. In the author's youth (he no longer enters competitions
nor judges them, nor does he know what happens nowadays)8 this remark
gave rise to abundantly exploited traps. Consider for example the sequence
8 The university examinations suffice. Peano once said that "relations between
students and professors would be excellent if it were not for the examinations,
which force students to consider their masters as potential judges" .