# 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 