207. The general power $a^x$

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 $$q\log y=p\log a,\log y=(p/q)\log a=x\log a,$$ 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 $a^x\to \mathrm{\infty }$ as $x\to \mathrm{\infty }$, 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 $x\to \mathrm{\infty }$, 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 $$D_a{a}^x=D_a{e}^{x\log a}=e^{x\log a}(x/a)=x{a}^{x-1}.$$

(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.