Lecture 5: Exponential Functions

$f(x) = a^x$ $\rightarrow a > 0, a \neq 1$

$D = \mathbb{R}, R = (0, \infty)$

*$a$ cannot be negative because if $x$ is a fraction, you can't root a negative #

^if $x = 0, a^0 = 1$

if $x = -n, {a^{-n}} = \frac{1}{a^n}$

if $x = \frac{1}{n}, {a^{\frac{1}{n}}} = \sqrt[n]{a}$

if $x = \frac{p}{q} = {a^{\frac{p}{q}}} = \sqrt[q]{a^p}$

1. $x^a x^b = x^{a + b}$
2. $\frac{x^a}{x^b} = x^{a - b}$
3. $(x^a)^b = x^{ab}$
4. $a^x \cdot b^x = (a \cdot b)^x$ or $\frac{a^x}{b^x} = (\frac{a}{b})^x$

*if $a > 0$ and $b > 0$ applies to all real numbers $x$ & $y$

Exponential Function $\rightarrow$ $f(x) = 10^x$

Power Function $\rightarrow g(x) = x^{10}$

Interest Example: In the year 2000, you invest $100 into a savings account w/ 5.5% interest compound annually. Find the formula for amount of $ in $x$ years.

Natural Exponential Functions $e^x$

$f(x) = e^x$ where $e = 2.718$... (an irrational #)

$D = \mathbb{R}$ & $R = (0, \infty)$

Exponential Growth & Decay

$f(x) = Q_0 e^{Kx}$ where $Q_0$ is a constant and if $K > 0$ = growth, $K < 0$ = decay