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)
$a^{-n} = \frac{1}{a^n}$
$a^{\frac{m}{n}} = \sqrt[n]{a^m}$

$a^x a^y = a^{x+y}$
$\frac{a^x}{a^y} = a^{x-y}$
$(ab)^x = a^x b^x$
$(a^x)^y = a^{xy}$

EXAMPLE 1

$5^{-2} = \frac{1}{5^2} = \frac{1}{25}$
$\left(\frac{3}{5}\right)^{-2} = \left(\frac{5}{3}\right)^2 = \frac{25}{9}$
$8^{\frac{2}{3}} = \sqrt[3]{8^2} = \sqrt[3]{64} = 4$ or $(2^3)^{\frac{2}{3}} = 2^2 = 4$
$27^{-4/3} = \sqrt[3]{27^{-4}} = \left(\frac{1}{3}\right)^4 = \frac{1}{81}$

2. The Exponential Function $f(x) = a^x$

Positivity: $f(x) = a^x$ evaluates strictly positive universally, thus the range is $y > 0$. Domain is unbounded: $x \in \mathbb{R}$.
Monotony: If $a > 1$, the function strictly increases. If $a < 1$, the function strictly decreases. If $a = 1$, it is flat constant.
Reflection: Applying $g(x) = \left(\frac{1}{a}\right)^x = a^{-x}$ reflects $a^x$ cleanly across the y-axis.
One-to-One Nature: Due to continuous strict monotony ($a \ne 1$), $a^x = a^y \Rightarrow x = y$. This resolves exponential algebraic equations.

EXAMPLE 2 (Resolving Exponential Equations)

Equations are solved algebraically by forcing matching basis variables on both operational sides:

(a) $2^{3x-1} = 2^{x+2} \Rightarrow 3x-1 = x+2 \Rightarrow 2x = 3 \Rightarrow x = 3/2$

(b) $2^{3x-1} = 4^{x+2} \Rightarrow 2^{3x-1} = 2^{2x+4} \Rightarrow 3x-1 = 2x+4 \Rightarrow x = 5$

(d) $\frac{1}{2^{3x-1}} = 4^{x+2} \Rightarrow 2^{-3x+1} = 2^{2x+4} \Rightarrow -3x+1 = 2x+4 \Rightarrow x = -3/5$

(e) $\sqrt{2}^{3x-1} = 4^{x+2} \Rightarrow 2^{\frac{3x-1}{2}} = 2^{2x+4} \Rightarrow \frac{3x-1}{2} = 2x+4 \Rightarrow x = -9$

3. The Natural Number $e$ and Asymptotes

A continuous mathematical limit evaluation defines the naturally occurring irrational base $e \approx 2.71828...$ Evaluating $\left(1+\frac{1}{n}\right)^n$ as $n \rightarrow \infty$ perfectly dictates this mathematical boundary.

Asymptotes: Graphically, exponential functions inherently possess horizontal asymptotes based exclusively on their applied vertical boundary shifts. $y=a^x + c$ establishes an exact HA bounding line at $y=c$.

EXAMPLE 3 (Horizontal Asymptote Variations)

Function	Horizontal Asymptote	y-intercept ($x=0$)
$f(x) = 2^x$	$y = 0$	$y = 1$
$f(x) = e^{3x}$	$y = 0$	$y = 1$
$f(x) = -3e^x$	$y = 0$	$y = -3$
$f(x) = e^x + 5$	$y = 5$	$y = 6$
$f(x) = 3e^x + 5$	$y = 5$	$y = 8$