The function $$a^{x}$$ has been defined only for rational values of $$x$$, except in the particular case when $$a = e$$. We shall now consider the case in which $$a$$ is any positive number. Suppose that $$x$$ is a positive rational number $$p/q$$. Then the positive value $$y$$ of the power $$a^{p/q}$$ is given by $$y^{q} = a^{p}$$; from which it follows that $q\log y = p\log a,\quad \log y = (p/q)\log a = x\log a,$ and so $y = e^{x\log a}.$ We take this as our definition of $$a^{x}$$ when $$x$$ is irrational. Thus $$10^{\sqrt{2}} = e^{\sqrt{2}\log 10}$$. It is to be observed that $$a^{x}$$, when $$x$$ is irrational, is defined only for positive values of $$a$$, and is itself essentially positive; and that $$\log a^{x} = x\log a$$. The most important properties of the function $$a^{x}$$ are as follows.

(1) Whatever value $$a$$ may have, $$a^{x} \times a^{y} = a^{x+y}$$ and $$(a^{x})^{y} = a^{xy}$$. In other words the laws of indices hold for irrational no less than for rational indices. For, in the first place, $a^{x} \times a^{y} = e^{x\log a} \times e^{y\log a} = e^{(x+y)\log a} = a^{x+y};$ and in the second $(a^{x})^{y} = e^{y\log a^{x}} = e^{xy\log a} = a^{xy}.$

(2) If $$a > 1$$ then $$a^{x} = e^{x\log a} = e^{\alpha x}$$, where $$\alpha$$ is positive. The graph of $$a^{x}$$ is in this case similar to that of $$e^{x}$$, and $$a^{x} \to \infty$$ as $$x \to \infty$$, more rapidly than any power of $$x$$.

If $$a < 1$$ then $$a^{x} = e^{x\log a} = e^{-\beta x}$$, where $$\beta$$ is positive. The graph of $$a^{x}$$ is then similar in shape to that of $$e^{x}$$, but reversed as regards right and left, and $$a^{x} \to 0$$ as $$x \to \infty$$, more rapidly than any power of $$1/x$$.

(3) $$a^{x}$$ is a continuous function of $$x$$, and $D_{x} a^{x} = D_{x} e^{x\log a} = e^{x\log a} \log a = a^{x} \log a.$

(4) $$a^{x}$$ is also a continuous function of $$a$$, and $D_{a} a^{x} = D_{a} e^{x\log a} = e^{x\log a} (x/a) = xa^{x-1}.$

(5) $$(a^{x} – 1)/x \to \log a$$ as $$x \to 0$$. This of course is a mere corollary from the fact that $$D_{x}a^{x} = a^{x}\log a$$, but the particular form of the result is often useful; it is of course equivalent to the result (Ex. LXXXV. 3) that $$(e^{\alpha x} – 1)/x \to \alpha$$ as $$x \to 0$$.

In the course of the preceding chapters a great many results involving the function $$a^{x}$$ have been stated with the limitation that $$x$$ is rational. The definition and theorems given in this section enable us to remove this restriction.