204–206. The exponential function

204. The exponential function.

We now define the exponential function \(e^{y}\) for all real values of \(y\) as the inverse of the logarithmic function. In other words we write \[x = e^{y}\] if \(y = \log x\).

We saw that, as \(x\) varies from \(0\) towards \(\infty\), \(y\) increases steadily, in the stricter sense, from \(-\infty\) towards \(\infty\). Thus to one value of \(x\) corresponds one value of \(y\), and conversely. Also \(y\) is a continuous function of \(x\), and it follows from § 109 that \(x\) is likewise a continuous function of \(y\).

It is easy to give a direct proof of the continuity of the exponential function. For if \(x = e^{y}\) and \(x + \xi = e^{y+\eta}\) then \[\eta = \int_{x}^{x+\xi} \frac{dt}{t}.\] Thus \(|\eta|\) is greater than \(\xi/(x + \xi)\) if \(\xi > 0\), and than \(|\xi|/x\) if \(\xi < 0\); and if \(\eta\) is very small \(\xi\) must also be very small.

Thus \(e^{y}\) is a positive and continuous function of \(y\) which increases steadily from \(0\) towards \(\infty\) as \(y\) increases from \(-\infty\) towards \(\infty\). Moreover \(e^{y}\) is the positive \(y\)th power of the number \(e\), in accordance with the elementary definitions, whenever \(y\) is a rational number. In particular \(e^{y} = 1\) when \(y = 0\). The general form of the graph of \(e^{y}\) is as shown in Fig. 53.

205. The principal properties of the exponential function.

(1) If \(x = e^{y}\), so that \(y = \log x\), then \(dy/dx = 1/x\) and \[\frac{dx}{dy} = x = e^{y}.\] Thus the derivative of the exponential function is equal to the function itself. More generally, if \(x = e^{ay}\) then \(dx/dy = ae^{ay}\).

(2) The exponential function satisfies the functional equation \[f(y + z) = f(y)f(z).\]

This follows, when \(y\) and \(z\) are rational, from the ordinary rules of indices. If \(y\) or \(z\), or both, are irrational then we can choose two sequences \(y_{1}\), \(y_{2}\), …, \(y_{n}\), … and \(z_{1}\), \(z_{2}\), …, \(z_{n}\), … of rational numbers such that \(\lim y_{n} = y\), \(\lim z_{n} = z\). Then, since the exponential function is continuous, we have \[e^{y} \times e^{z} = \lim e^{y_{n}} \times \lim e^{z_{n}} = \lim e^{y_{n}+z_{n}} = e^{y+z}.\] In particular \(e^{y} \times e^{-y} = e^{0} = 1\), or \(e^{-y} = 1/e^{y}\).

We may also deduce the functional equation satisfied by \(e^{y}\) from that satisfied by \(\log x\). For if \(y_{1} = \log x_{1}\), \(y_{2} = \log x_{2}\), so that \(x_{1} = e^{y_{1}}\), \(x_{2} = e^{y_{2}}\), then \(y_{1} + y_{2} = \log x_{1} + \log x_{2} = \log x_{1}x_{2}\) and \[e^{y_{1}+y_{2}} = e^{\log x_{1}x_{2}} = x_{1}x_{2} = e^{y_{1}} \times e^{y_{2}}.\]

206.

(3) The function \(e^{y}\) tends to infinity with \(y\) more rapidly than any power of \(y\), or \[\lim y^{\alpha}/e^{y} = \lim e^{-y}y^{\alpha} = 0\] as \(y \to \infty\), for all values of \(\alpha\) however great.

We saw that \((\log x)/x^{\beta} \to 0\) as \(x \to \infty\), for any positive value of \(\beta\) however small. Writing \(\alpha\) for \(1/\beta\), we see that \((\log x)^{\alpha}/x \to 0\) for any value of \(\alpha\) however large. The result follows on putting \(x = e^{y}\). It is clear also that \(e^{\gamma y}\) tends to \(\infty\) if \(\gamma > 0\), and to \(0\) if \(\gamma < 0\), and in each case more rapidly than any power of \(y\).

From this result it follows that we can construct a ‘scale of infinity’ similar to that constructed in § 202, but extending in the opposite direction; i.e. a scale of functions which tend to \(\infty\) more and more rapidly as \(x \to \infty\).¹ The scale is \[x,\quad x^{2},\quad x^{3},\ \dots\quad e^{x},\quad e^{2x},\ \dots\quad e^{x^{2}},\ \dots,\quad e^{x^{3}},\ \dots,\quad e^{e^{x}},\ \dots,\] where of course \(e^{x^{2}}\), …, \(e^{e^{x}}\), … denote \(e^{(x^{2})}\), …, \(e^{(e^{x})}\), ….

The reader should try to apply the remarks about the logarithmic scale, made in § 202 and Ex. LXXXIV, to this ‘exponential scale’ also. The two scales may of course (if the order of one is reversed) be combined into one scale \[\dots\ \log\log x,\ \dots\quad \log x,\ \dots\quad x,\ \dots\quad e^{x},\ \dots\quad e^{e^{x}},\ \dots.\]