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 $$e < 3$$.]