## 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 $$0 < |\eta| < |y|$$.]

## 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 $\dots\ \log\log x,\ \dots\quad \log x,\ \dots\quad x,\ \dots\quad e^{x},\ \dots\quad e^{e^{x}},\ \dots.$

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.↩︎