203. The number $e$

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 $$\log x^2=2\log x,\log x^3=3\log x,\dots ,\log x^n=n\log x,$$ 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 $$\begin{array}{}\text{(1)}& \log e^y=y,\end{array}$$ 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 $$\begin{array}{}\text{(2)}& y=\log x,x=e^y\end{array}$$ 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 $e<3$.]