We shall now give an outline of a method of investigation of the properties of \(e^{x}\) and \(\log x\) entirely different in logical order from that followed in the preceding pages. This method starts from the exponential series \(1 + x + \dfrac{x^{2}}{2!} + \dots\). We know that this series is convergent for all values of \(x\), and we may therefore define the function \(\exp x\) by the equation \[\begin{equation*} \exp x = 1 + x + \frac{x^{2}}{2!} + \dots. \tag{1} \end{equation*}\]

We then prove, as in Ex. LXXXI. 7, that \[\begin{equation*} \exp x \times \exp y = \exp(x + y). \tag{2} \end{equation*}\]

Again \[\frac{\exp h – 1}{h} = 1 + \frac{h}{2!} + \frac{h^{2}}{3!} + \dots = 1 + \rho(h),\] where \(\rho(h)\) is numerically less than \[|\tfrac{1}{2}h| + |\tfrac{1}{2}h|^{2} + |\tfrac{1}{2}h|^{3} + \dots = |\tfrac{1}{2}h|/(1 – |\tfrac{1}{2}h|),\] so that \(\rho(h) \to 0\) as \(h \to 0\). And so \[\frac{\exp(x + h) – \exp x}{h} = \exp x \left(\frac{\exp h – 1}{h}\right) \to \exp x\] as \(h \to 0\), or \[\begin{equation*} D_{x} \exp x = \exp x. \tag{3} \end{equation*}\] Incidentally we have proved that \(\exp x\) is a continuous function.

We have now a choice of procedure. Writing \(y = \exp x\) and observing that \(\exp 0 = 1\), we have \[\frac{dy}{dx} = y,\quad x = \int_{1}^{y} \frac{dt}{t},\] and, if we define the logarithmic function as the function inverse to the exponential function, we are brought back to the point of view adopted earlier in this chapter.

But we may proceed differently. From it follows that if \(n\) is a positive integer then \[(\exp x)^{n} = \exp nx,\quad (\exp 1)^{n} = \exp n.\] If \(x\) is a positive rational fraction \(m/n\), then \[\{\exp(m/n)\}^{n} = \exp m = (\exp 1)^{m},\] and so \(\exp(m/n)\) is equal to the positive value of \((\exp 1)^{m/n}\). This result may be extended to negative rational values of \(x\) by means of the equation \[\exp x \exp(-x) = 1;\] and so we have \[\exp x = (\exp 1)^{x} = e^{x},\] say, where \[e = \exp 1 = 1 + 1 + \frac{1}{2!} + \frac{1}{3!} + \dots,\] for all rational values of \(x\). Finally we define \(e^{x}\), when \(x\) is irrational, as being equal to \(\exp x\). The logarithm is then defined as the function inverse to \(\exp x\) or \(e^{x}\).

*Example.* Develop the theory of the binomial series \[1 + \binom{m}{1} x + \binom{m}{2} x^{2} + \dots = f(m, x),\] where \(-1 < x < 1\), in a similar manner, starting from the equation \[f(m, x) f(m’, x) = f(m + m'{,} x)\] (Ex. LXXXI. 6).

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