Lesson 2: The Natural Base e

Property

The number $e$ is an irrational number approximately equal to $2.71828...$. It is defined as the value that the expression $(1 + \frac{1}{n})^n$ approaches as $n$ gets infinitely large. The function $f(x) = e^x$ is called the natural exponential function.

Examples

• The value of the natural exponential function at $x = 3$ is $f(3) = e^3 \approx (2.718)^3 \approx 20.086$.

• A population of cells growing continuously can be modeled by $P(t) = 50e^{0.1t}$, where 50 is the initial number of cells.

Property

Natural Base e: The number $e$ is defined as the value of $(1 + \frac{1}{n})^n$, as $n$ increases without bound. We say, as $n$ approaches infinity,
$e \approx 2.718281827\ldots$

Natural Exponential Function: The natural exponential function is an exponential function whose base is $e$

The domain is $(-\infty, \infty)$ and the range is $(0, \infty)$.

Examples

• The graph of the natural exponential function $f(x) = e^x$ is an increasing curve that sits between the graphs of $g(x) = 2^x$ and $h(x) = 3^x$, since $2 < e < 3$.
• As you calculate $(1 + \frac{1}{n})^n$ for larger and larger values of $n$, the result gets closer and closer to $e \approx 2.718281827$.
• Using a calculator, the value of $e^2$ is approximately $7.389$. This corresponds to the point $(2, 7.389)$ on the graph of $f(x)=e^x$.