Definition 1

Tag 089S, permanent. Cite it as [Bourbaki, Tag 089S].

Functions of a Real Variable, chapter III, § 1. DERIVATIVES OF THE EXPONENTIAL AND CIRCULAR FUNCTIONS

Logarithms to the base e are called Naperian logarithms (or natural logarithms).

We usually omit the base in the notation for the Naperian logarithm. Unless it is stated to the contrary, the notation $\log x$ ( $x > 0$ ) will denote the Naperian logarithm of $x$ . With this notation, prop. 1 can be written as the identity

D(a^\lambda) = (\log a)a^\lambda

(4)

valid for any $a > 0$ ( $\log a = 0$ when $a = 1$ ).

This relation shows that $a^x$ has derivatives of every order, and that

D^n(a^x) = (\log a)^n a^x.

(5)

In particular, for $a > 0$ and $\neq 1$ one has $D^2(a^x) > 0$ for all $x \in \mathbf{R}$ , and hence $a^x$ is strictly convex on $\mathbf{R}$ (I, p. 31, corollary). From this one deduces the following proposition:

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

Cited by

Nothing.

Cites

I, p. 31