6.3 The Natural Exponential
An exponential function allows its exponent to vary.
Definition.
An exponential function is a power
\[
\begin{align}
p &\depends e
\\
p &= b^e
\end{align}
\]
where the exponent, \( e \), is allowed to vary and the base, \( b \), is constant.
The natural exponential function is the exponential function
\[
\begin{align}
p &\mathrel{\overset{\exp}{\leftarrow}} e
\\
p &= \naturalbase^e
\end{align}
\]
where the constant
\[
\begin{align}
\naturalbase &: \Number
\\
\naturalbase &= 1 + \frac{1}{1} + \frac{1}{1 \mult 2} + \frac{1}{1 \mult 2 \mult 3} + \frac{1}{1 \mult 2 \mult 3 \mult 4} + \cdots
\\
\naturalbase &= 2.718 \ldots
\end{align}
\]
is an irrational number is known as the natural base. Don't worry about the natural base's strange definition: it was carefully chosen to give us simple differential laws. The constant \( \naturalbase \) is just some number between two and three that we call "natural."
Theorem.
The natural exponential function satisfies the laws
\[
\begin{align}
&\exp 0 = 1
\\
&\exp 1 = \naturalbase
\\
&\exp (e_1 + e_2) = \exp e_1 \mult \exp e_2
\end{align}
\]
where \( e_1 : \Number \) and \( e_2 : \Number \) are numbers.
Why?
These follow quickly from our definition of the natural exponential.
\[
\begin{align}
&\exp 0 = \naturalbase^0 = 1
\\
&\exp 1 = \naturalbase^1 = \naturalbase
\\
&\exp (e_1 + e_2) = \naturalbase^{e_1 + e_2} = \naturalbase^{e_1} \mult \naturalbase^{e_2} = \exp e_1 \mult \exp e_2
\end{align}
\]
Graphing the natural exponential function by hand takes some real effort. We can quickly plot a few points if we use a calculator or computer.
| \( e \) | \( p \) |
|---|---|
| \( -2 \) | \( \approx 0.135 \) |
| \( -1 \) | \( \approx 0.367 \) |
| \( 0 \) | \( 1 \) |
| \( 1 \) | \( \approx 2.718 \) |
| \( 2 \) | \( \approx 7.398 \) |
To gain more familiarity with exponential functions, you can graph the base two exponential, \( 2^e \). This function is much easier to graph, and it has a very similar shape. In fact, the graphs of exponential functions are all quite similar. We'll soon see a change-of-base formula which reveals an important geometric fact: exponential functions only differ by horizontal transformers.