The Natural Exponential Function

Definition The number $e$ is the exact real constant

where $n$ runs through the positive integers $1, 2, 3, \dots$ It is irrational, so its decimal expansion never ends and never repeats. The decimal $2.71828$ is an approximation of $e$, not its definition. Here we take on trust that this limit exists. The proof, and the property of $e$ that makes it special, come in calculus.

Definition The natural exponential function is

Its base $e$ is positive and not equal to $1$, so it belongs to the family $a^x$ with $a > 0$, $a \ne 1$. Because $e > 1$, the function is strictly increasing on $\mathbb{R}$, and $e^x > 0$ for every real $x$. Its range is $(0, \infty)$. For all real $x$ and $y$,

Definition An exact value is written with symbols that name the number precisely, such as $e^2$, $\frac{1}{e}$ or $3e^{0.5}$. An approximate value is a decimal rounded to some number of places, such as $7.389$, written with $\approx$ rather than $=$. Rounding intermediate results changes later results, so a careful computation keeps exact values until the last step.

Definition For any fixed increment $\Delta$ and every real $x$,

The same holds for $f(t) = C e^{kt}$ with $C \ne 0$, where

The factor depends only on the increment, not on the starting input.

Definition The function $e^{-x} = \left(\frac{1}{e}\right)^x$ has base $\frac{1}{e}$, which lies between $0$ and $1$, so it is strictly decreasing on $\mathbb{R}$. A model

describes exponential decay. It starts at $Q(0) = Q_0$ and keeps the fraction $e^{-k\Delta}$ of its value over every input step $\Delta$.

Definition As $x \to -\infty$ the values $e^x$ approach $0$, and as $x \to \infty$ the values $e^{-x}$ approach $0$. So $y = 0$ is a horizontal asymptote of both graphs, on one side each, and neither graph reaches it. For $f(x) = C e^{kx} + L$ with $C \ne 0$ and $k \ne 0$, the horizontal asymptote is $y = L$.

Definition A scaled exponential is a function

Since $e^{kx} = \left(e^k\right)^x$, it is an exponential function with base $e^k$. For $k > 0$ the base is greater than $1$, and for $k < 0$ it lies between $0$ and $1$. The graph of $y = e^{kx}$ is the graph of $y = e^x$ scaled horizontally by the factor $\frac{1}{\lvert k \rvert}$ toward or away from the $y$-axis, a compression when $\lvert k \rvert > 1$ and a stretch when $\lvert k \rvert < 1$, and also reflected across the $y$-axis when $k < 0$. Conversely, every base $a > 0$ with $a \ne 1$ equals $e^k$ for exactly one real number $k \ne 0$, so every exponential function can be written with base $e$. That $k$ is the natural logarithm of $a$, which the natural logarithm lesson introduces.