2.8 Exponents: The Exponential Function $a^x$
1. Definition and Algebraic Properties
Power evaluations define numerical growth across increasingly expansive number sets (Natural $\mathbb{N}$, Integer $\mathbb{Z}$, Rational $\mathbb{Q}$, Real $\mathbb{R}$). For any base positive integer $a > 0$:
- $a^0 = 1$
- $a^n = a \cdot a \cdot \dots \cdot a$ ($n$ times)
- $a^{-n} = \dfrac{1}{a^n}$
- $a^{\frac{m}{n}} = \sqrt[n]{a^m}$
- $a^x a^y = a^{x+y}$
- $\dfrac{a^x}{a^y} = a^{x-y}$
- $(ab)^x = a^x b^x$
- $(a^x)^y = a^{xy}$
NOTICE
- If $a < 0$, then $a^x$ is generally defined only for integer exponents $x = n \in \mathbb{Z}$.
- $0^x = 0$ only when $x > 0$.
- $0^0$ is not defined.
EXAMPLE 1
- $5^{-2} = \dfrac{1}{5^2} = \dfrac{1}{25}$
- $\left(\dfrac{3}{5}\right)^{-2} = \left(\dfrac{5}{3}\right)^2 = \dfrac{25}{9}$
- $8^{\frac{2}{3}} = \sqrt[3]{8^2} = \sqrt[3]{64} = 4 \quad \text{or} \quad (2^3)^{\frac{2}{3}} = 2^2 = 4$
- $27^{-4/3} = \sqrt[3]{27^{-4}} = \left(\dfrac{1}{3}\right)^4 = \dfrac{1}{81}$
2. The Exponential Function $f(x) = a^x$ where $a > 0$
Consider $f:\mathbb{R} \rightarrow \mathbb{R}, \quad f(x) = 2^x$. The smooth graph of the exponential function is drawn first:
| $x$ | $\cdots$ | $-3$ | $-2$ | $-1$ | $0$ | $1$ | $2$ | $3$ | $\cdots$ |
|---|---|---|---|---|---|---|---|---|---|
| $y = 2^x$ | $\cdots$ | $\dfrac{1}{8}$ | $\dfrac{1}{4}$ | $\dfrac{1}{2}$ | $1$ | $2$ | $4$ | $8$ | $\cdots$ |
Consider now $g:\mathbb{R} \rightarrow \mathbb{R}, \quad g(x) = 0.5^x \quad \left[\text{that is } g(x) = \left(\dfrac{1}{2}\right)^x = \dfrac{1}{2^x}\right]$. Its smooth graph is drawn first:
| $x$ | $\cdots$ | $-3$ | $-2$ | $-1$ | $0$ | $1$ | $2$ | $3$ | $\cdots$ |
|---|---|---|---|---|---|---|---|---|---|
| $y = \left(\dfrac{1}{2}\right)^x$ | $\cdots$ | $8$ | $4$ | $2$ | $1$ | $\dfrac{1}{2}$ | $\dfrac{1}{4}$ | $\dfrac{1}{8}$ | $\cdots$ |
NOTICE
- $f(x)=a^x$ is always positive, even if $x < 0$.
- $g(x)=\left(\dfrac{1}{a}\right)^x = \dfrac{1}{a^x} = a^{-x}$. Thus, $g(x)$ is a reflection of $f(x)=a^x$ about the $y$-axis.
-
If $a > 1$, then $f(x)=a^x$ increases.
If $0 < a < 1$, then $f(x)=a^x$ decreases.
If $a = 1$, then $f(x)=1^x=1$ is constant. -
If $a \ne 1$, then $f(x)=a^x$ is one-to-one, i.e.
$a^x=a^y \Rightarrow x=y$
This property helps us solve exponential equations.
EXAMPLE 2 (Resolving Exponential Equations)
Equations are solved algebraically by forcing matching basis variables on both operational sides:
3. The Natural Number $e$ and Asymptotes
There is a specific irrational number
which plays an important role in mathematics. The number $e$ is almost as popular as the irrational number $\pi \approx 3.14\dots$
An approximation of $e$ is given by the expression
| For $n=1$ | the result is | $2$ |
| For $n=2$ | the result is | $2.25$ |
| For $n=10$ | the result is | $2.5937424\dots$ |
| For $n=100$ | the result is | $2.7048138\dots$ |
| For $n=1000$ | the result is | $2.7169239\dots$ |
| For $n=10^6$ | the result is | $2.7182804\dots$ |
As $n$ tends to $+\infty$, this expression tends to $e \approx 2.7182818\dots$
4. The Exponential $e^x$
The exponential function $f(x)=e^x$ appears in many applications. The graph looks like any function of the form $f(x)=a^x$ with $a > 1$.
5. Asymptotes of Exponential Functions
Observe the exponential functions $(a > 0, \ a \ne 1)$:
EXAMPLE 3
| Function | Horizontal Asymptote | y-intercept ($x=0$) |
|---|---|---|
| $f(x) = 2^x$ | line $y = 0$ | $y = 1$ |
| $f(x) = 2^{-x}$ | line $y = 0$ | $y = 1$ |
| $f(x) = e^x$ | line $y = 0$ | $y = 1$ |
| $f(x) = e^{3x}$ | line $y = 0$ | $y = 1$ |
| $f(x) = 3e^x$ | line $y = 0$ | $y = 3$ |
| $f(x) = -3e^x$ | line $y = 0$ | $y = -3$ |
| $f(x) = e^x + 5$ | line $y = 5$ | $y = 6$ |
| $f(x) = 3e^x + 5$ | line $y = 5$ | $y = 8$ |
| $f(x) = e^{x-2}$ | line $y = 0$ | $y = e^{-2}$ |