We shall now introduce a number, usually denoted by \(e\), which is of immense importance in higher mathematics. It is, like \(\pi\), one of the fundamental constants of analysis.

We define \(e\) as *the number whose logarithm is \(1\)*. In other words \(e\) is defined by the equation \[1 = \int_{1}^{e} \frac{dt}{t}.\] Since \(\log x\) is an increasing function of \(x\), in the stricter sense of § 95, it can only pass once through the value \(1\). Hence our definition does in fact define one definite number.

Now \(\log xy = \log x + \log y\) and so \[\log x^{2} = 2\log x,\quad \log x^{3} = 3\log x,\ \dots,\quad \log x^{n} = n\log x,\] where \(n\) is any positive integer. Hence \[\log e^{n} = n\log e = n.\] Again, if \(p\) and \(q\) are any positive integers, and \(e^{p/q}\) denotes the positive \(q\)th root of \(e^{p}\), we have \[p = \log e^{p} = \log(e^{p/q})^{q} = q\log e^{p/q},\] so that \(\log e^{p/q} = p/q\). Thus, if \(y\) has any positive rational value, and \(e^{y}\) denotes the positive \(y\)th power of \(e\), we have \[\begin{equation*} \log e^{y} = y, \tag{1} \end{equation*}\] and \(\log e^{-y} = -\log e^{y} = -y\). Hence the equation (1) is true for all rational values of \(y\), positive or negative. In other words the equations \[\begin{equation*} y = \log x,\quad x = e^{y} \tag{2} \end{equation*}\] are consequences of one another so long as \(y\) is rational and \(e^{y}\) has its positive value. At present we have not given any definition of a power such as \(e^{y}\) in which the index is irrational, and the function \(e^{y}\) is defined for rational values of \(y\) only.

*Example.* Prove that \(2 < e < 3\). [In the first place it is evident that \[\int_{1}^{2} \frac{dt}{t} < 1,\] and so \(2 < e\). Also \[\int_{1}^{3} \frac{dt}{t} = \int_{1}^{2} \frac{dt}{t} + \int_{2}^{3} \frac{dt}{t} = \int_{0}^{1} \frac{du}{2 – u} + \int_{0}^{1} \frac{du}{2 + u} = 4\int_{0}^{1} \frac{du}{4 – u^{2}} > 1,\] so that \(e < 3\).]

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