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 \[q\log y = p\log a,\quad \log y = (p/q)\log a = x\log a,\] 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.