Defining the Exponential Function — Cheat sheet

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Definition

$f(x) = a^x$ , with $a > 0$ and $a \ne 1$ .

The base is fixed; the variable is the power. That is the whole difference from $x^3$ .

Expression	Base	Power
$x^3$	varies	fixed
$3^x$	fixed	varies

Condition	What it prevents
$a > 0$	$a^{1/2} = \sqrt{a}$ has no real value for a negative base
$a \ne 1$	$1^x = 1$ is a flat line, with no inverse

$a^x \cdot a^y = a^{x + y}$ , $\dfrac{a^x}{a^y} = a^{x - y}$ , $\left(a^x\right)^y = a^{xy}$

$a^0 = 1$ , and $a^{-x} = \dfrac{1}{a^x}$ - a negative power is a reciprocal, never a negative value.

Fact	Consequence
$a^x > 0$ always	the curve never touches the $x$ -axis
$a^0 = 1$	every exponential passes through $(0, 1)$
$a > 1$	growth: climbs without limit
$0 < a < 1$	decay: falls towards $0$

Put both sides on the same base, then equate the powers. An exponential takes each value once, so nothing is lost.