Powers

Section A.12 Powers

In the following,

x

and

y

are arbitrary real numbers, and

q

is an arbitrary constant that is strictly bigger than zero.

$q^{0} = 1$
$q^{x + y} = q^{x} q^{y},$ $q^{x - y} = \frac{q^{x}}{q^{y}}$
$q^{- x} = \frac{1}{q^{x}}$
$(q^{x})^{y} = q^{x y}$
$lim_{x \to \infty} q^{x} = \infty,$ $lim_{x \to - \infty} q^{x} = 0$ if $q > 1$
$lim_{x \to \infty} q^{x} = 0,$ $lim_{x \to - \infty} q^{x} = \infty$ if $0 < q < 1$
The graph of $2^{x}$ is given below. The graph of $q^{x},$ for any $q > 1,$ is similar.