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 \[e^{y_{1}+y_{2}} = e^{\log x_{1}x_{2}} = x_{1}x_{2} = e^{y_{1}} \times e^{y_{2}}.\]

Example LXXXV
1. If \(dx/dy = ax\) then \(x = Ke^{ay}\), where \(K\) is a constant.

2. There is no solution of the equation \(f(y + z) = f(y)f(z)\) fundamentally distinct from the exponential function. [We assume that \(f(y)\) has a differential coefficient. Differentiating the equation with respect to \(y\) and \(z\) in turn, we obtain \[f'(y + z) = f'(y)f(z),\quad f'(y + z) = f(y)f'(z)\] and so \(f'(y)/f(y) = f'(z)/f(z)\), and therefore each is constant. Thus if \(x = f(y)\) then \(dx/dy = ax\), where \(a\) is a constant, so that \(x = Ke^{ay}\) (Ex. 1).]

3. Prove that \((e^{ay} – 1)/y \to a\) as \(y \to 0\). [Applying the Mean Value Theorem, we obtain \(e^{ay} – 1 = aye^{a\eta}\), where \(0 < |\eta| < |y|\).]



(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 \[\dots\ \log\log x,\ \dots\quad \log x,\ \dots\quad x,\ \dots\quad e^{x},\ \dots\quad e^{e^{x}},\ \dots.\]

  1. The exponential function was introduced by inverting the equation \(y = \log x\) into \(x = e^{y}\); and we have accordingly, up to the present, used \(y\) as the independent and \(x\) as the dependent variable in discussing its properties. We shall now revert to the more natural plan of taking \(x\) as the independent variable, except when it is necessary to consider a pair of equations of the type \(y = \log x\), \(x = e^{y}\) simultaneously, or when there is some other special reason to the contrary.↩︎

$\leftarrow$203. The number \(e\) Main Page 207. The general power \(a^{x}\) $\rightarrow$