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 $x + ξ = e^{y + η}$ then $η = \int_{x}^{x + ξ} \frac{d t}{t} .$ Thus $| η |$ is greater than $ξ / (x + ξ)$ if $ξ > 0$ , and than $| ξ | / x$ if $ξ < 0$ ; and if $η$ is very small $ξ$ 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 $d y / d x = 1 / x$ and $\frac{d x}{d y} = x = e^{y} .$ Thus the derivative of the exponential function is equal to the function itself. More generally, if $x = e^{a y}$ then $d x / d y = a e^{a y}$ .

(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

d x / d y = a x

then

x = K e^{a y}

, 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 $f^{'} (y + z) = f^{'} (y) f (z), f^{'} (y + z) = f (y) f^{'} (z)$ and so $f^{'} (y) / f (y) = f^{'} (z) / f (z)$ , and therefore each is constant. Thus if $x = f (y)$ then $d x / d y = a x$ , where $a$ is a constant, so that $x = K e^{a y}$ (Ex. 1).]

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

206.

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

We saw that $(\log x) / x^{β} \to 0$ as $x \to \infty$ , for any positive value of $β$ however small. Writing $α$ for $1 / β$ , we see that $(\log x)^{α} / x \to 0$ for any value of $α$ however large. The result follows on putting $x = e^{y}$ . It is clear also that $e^{γ y}$ tends to $\infty$ if $γ > 0$ , and to $0$ if $γ < 0$ , 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 $x, x^{2}, x^{3}, \dots e^{x}, e^{2 x}, \dots e^{x^{2}}, \dots, e^{x^{3}}, \dots, e^{e^{x}}, \dots,$ 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 $\dots \log \log x, \dots \log x, \dots x, \dots e^{x}, \dots e^{e^{x}}, \dots .$

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.↩︎

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203. The number

e

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207. The general power

a^{x}

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