Page:Calculus Made Easy.pdf/165

and therefore

when $x=x,\quad y=(2.718281{\text{ etc}}.)^{x}$; that is, $y=\epsilon ^{x},$ thus finally demonstrating that

$\epsilon ^{x}=1+{\dfrac {x}{1}}+{\dfrac {x^{2}}{1\cdot 2}}+{\dfrac {x^{3}}{1\cdot 2\cdot 3}}+{\dfrac {x^{4}}{1\cdot 2\cdot 3\cdot 4}}+{\text{etc}}.$

[Note.–How to read exponentials. For the benefit of those who have no tutor at hand it may be of use to state that $\epsilon ^{x}$ is read as “epsilon to the eksth power;” or some people read it “exponential eks.” So $\epsilon ^{pt}$ is read “epsilon to the pee-teeth-power” or “exponential pee tee.” Take some similar expressions:–Thus, $\epsilon ^{-2}$ is read “epsilon to the minus two power” or “exponential minus two.” $\epsilon ^{-ax}$ is read “epsilon to the minus ay-eksth” or “exponential minus ay-eks.”]

Of course it follows that $\epsilon ^{y}$ remains unchanged if differentiated with respect to $y$. Also $\epsilon ^{ax}$, which is equal to $(\epsilon a)^{x}$, will, when differentiated with respect to $x$, be $a\epsilon ^{ax}$, because a is $a$ constant.

Natural or Naperian Logarithms.

Another reason why $\epsilon$ is important is because it was made by Napier, the inventor of logarithms, the basis of his system. If $y$ is the value of $\epsilon ^{x}$, then $x$ is the logarithm, to the base $\epsilon$, of $y$. Or, if

${\begin{aligned}y&=\epsilon ^{x},\\{\text{then}}\;x&=\log _{\epsilon }y.\end{aligned}}$

The two curves plotted in Fig. 38 and 39 represent these equations.