Natural Exponential


We learnt that the simplest exponential functions are of the form $y=a^x$ where $a>0$, $a \ne 1$.
We can see that for all positive values of the base $a$, the graph is always positive, that is $a^x > 0$ for all $a>0$.
There are an infinite number of possible choices for the base number.

However, where exponential data is examined in engineering, science, and finance, the base $e = 2.7183 \cdots$ is commonly used.

$e$ is a special number in mathematics. It is irrational like $\pi$, and just as $\pi$ is the ratio of a circle’s circumference to its diameter, $e$
also has a physical meaning.

$\textit{Definition of e}$

$$e = \lim_{n \rightarrow \infty}\Big(1+\dfrac{1}{n}\Big)^n$$
\begin{array}{|c|c|} \hline
n & \Big(1+\dfrac{1}{n}\Big)^n \\ \hline
1 & \Big(1+\dfrac{1}{1}\Big)^1 = 2 \\ \hline
10 & \Big(1+\dfrac{1}{10}\Big)^{10} = 2.5937 \cdots \\ \hline
100 & \Big(1+\dfrac{1}{100}\Big)^{100} = 2.7048 \cdots \\ \hline
1000 & \Big(1+\dfrac{1}{1000}\Big)^{1000} = 2.7169 \cdots \\ \hline
10000 & \Big(1+\dfrac{1}{10000}\Big)^{10000} = 2.7181 \cdots \\ \hline
1000 \cdots 0 & \Big(1+\dfrac{1}{1000 \cdots 0}\Big)^{1000 \cdots 0} = 2.7183 \cdots \\ \hline
\end{array}
or
$$ \begin{align} \displaystyle
e &= \sum_{n=0}^{\infty}\dfrac{1}{n!} \\
&= \dfrac{1}{0!} + \dfrac{1}{1!} + \dfrac{1}{2!} + \dfrac{1}{3!} + \cdots \\
&= \dfrac{1}{1} + \dfrac{1}{1} + \dfrac{1}{1 \times 2} + \dfrac{1}{1 \times 2 \times 3} + \cdots \\
\end{align} $$

Example 1

Find $e^3$ to 3 significant figures.

\( \begin{align} \displaystyle
e^3 &= 20.08553 \cdots \\
&\approx 20.1 \\
\end{align} \)

Example 2

Find $e^{0.32}$ to 3 significant figures.

\( \begin{align} \displaystyle
e^{0.32} &= 1.377127 \cdots \\
&\approx 1.38 \\
\end{align} \)

Example 3

Find $\sqrt{e}$ to 3 significant figures.

\( \begin{align} \displaystyle
\sqrt{e} &= 1.64872 \cdots \\
&\approx 1.65 \\
\end{align} \)

Example 4

Find $e^{-2}$ to 3 significant figures.

\( \begin{align} \displaystyle
e^{-2} &= 0.135335 \cdots \\
&\approx 0.135 \\
\end{align} \)

Example 5

Write $\sqrt{e}$ as powers of $e$.

$\sqrt{e} = e^{\frac{1}{2}}$

Example 6

Write $\dfrac{1}{e^2}$ as powers of $e$.

$\dfrac{1}{e^2} = e^{-2}$

Example 7

Expand $(e^x + 2)^2$.

\( \begin{align} \displaystyle
(e^x + 2)^2 &= (e^x)^2 + 4e^x + 4 \\
&= e^{2x} + 4e^x + 4 \\
\end{align} \)

Example 8

Solve $e^x = \sqrt{e}$ for $x$.

\( \begin{align} \displaystyle
e^x &= e^{\frac{1}{2}} \\
x &= \dfrac{1}{2} \\
\end{align} \)


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