Theorem 1

The multiplicative group $\mathbf{R}_+^*$ of real numbers $> 0$ is a topological group isomorphic to the additive group $\mathbf{R}$ of real numbers.

For $\mathbf{R}_+^* = ]0, +\infty[$ is an open interval of $\mathbf{R}$ and is therefore homeomorphic to $\mathbf{R}$ (Chapter IV, § 4, no. 1, Proposition 1). By Theorem 2 of § 3, it is therefore a topological group isomorphic to $\mathbf{R}$.

From the Corollary to Proposition 5 of § 1, no. 3, for every number $a > 0$ there is a unique continuous homomorphism $f_a$ of $\mathbf{R}$ into $\mathbf{R}_+^*$ such that $f_a(1) = a$. Hence, for all $x \in \mathbf{R}$ and all $y \in \mathbf{R}$ we have

and hence, in particular, for all $n \in \mathbf{Z}$,

For this reason we denote $f_a(x)$ by $a^x$ for all $x \in \mathbf{R}$; the functions $a^x$ (for all values of $a > 0$) are called exponential functions. We have $1^x = 1$ for all $x \in \mathbf{R}$; if $a \neq 1$, $x \to a^x$ is an isomorphism of the group $\mathbf{R}$ onto the group $\mathbf{R}_+^*$.

If $a \neq 1$, the isomorphism of $\mathbf{R}_+^*$ onto $\mathbf{R}$ which is the inverse of $x \to a^x$ is called the logarithm to base $a$, and its value at $x \in \mathbf{R}_+^*$ is denoted by $\log_a x$. Thus, with this notation, we have

(1)

(2) (3) (4) (5) (6) (7)

By Proposition 5 of § 1, no. 3, every continuous homomorphism of $\mathbf{R}$ into $\mathbf{R}_+^*$ is of the form $y \to a^{xy}$, where $x \in \mathbf{R}$; since its value when $y = 1$ is $a^x$, we have identically

(8)

or, changing the notation,

(9)

The formula (8) shows that, for every integer $n > 0$, we have $(a^{1/n})^n = a$, which justifies the notation $a^{1/n}$ introduced for the nth root $\sqrt[n]{a}$, defined in Chapter IV, § 3, no. 3.

Formulas (7) and (9) show that

(10)

or, changing the notation,

(11)

which is the formula for "change of base".

Finally, let us obtain all the continuous homomorphisms of the topological group $\mathbf{R}^*$ into itself; if $g$ is such a continuous homomorphism,

is a continuous homomorphism of $\mathbf{R}$ into $\mathbf{R}$, therefore (§ 1, no. 3, Proposition 5) there exists $\alpha \in \mathbf{R}$ such that $\log_a (g(a^x)) = \alpha x$ for all $x \in \mathbf{R}$; hence, by (8), $g(x) = x^\alpha$ for all $x > 0$. Hence we have identically

(12)

By reason of formula (4), which reduces every multiplication to an addition (the only operation to which the customary system of numeration is well adapted), logarithms have long been an indispensable instrument in numerical calculations (see the Historical Note to this Chapter).

When used for this purpose, the base chosen is $a = 10$; and there are tables giving the values of the function $\log_{10} x$ (to a certain approximation). In analysis, one is led to choose a different base (denoted by $e$) which is such that $\lim_{x \to 1, x \neq 1} \frac{\log_e x}{(x - 1)} = 1$ (cf. Exercise 1).