# Math 22 Exponential Functions

## Definition of Exponential Function

 If $a>0$ and $a\neq 1$ , then the exponential function with base $a$ is $a^{x}$ ## Properties of Exponents

Let $a$ and $b$ be positive real numbers, and let $x$ and $y$ be real numbers.

1.$a^{0}=1$ 2.$a^{x}a^{y}=a^{x+y}$ 3.${\frac {a^{x}}{a^{y}}}=a^{x-y}$ 4.$(a^{x})^{y}=a^{xy}$ 5.$(ab)^{x}=a^{x}b^{x}$ 6.$({\frac {a}{b}})^{x}={\frac {a^{x}}{b^{x}}}$ 7.$a^{-x}={\frac {1}{a^{x}}}$ Exercises Use the properties of exponents to simplify each expression:

a) $(8^{\frac {1}{2}})(2^{\frac {1}{2}})$ Solution:
$(8^{\frac {1}{2}})(2^{\frac {1}{2}})=(8\cdot 2)^{\frac {1}{2}}=16^{\frac {1}{2}}$ b) ${\frac {7^{5}}{49^{3}}}$ Solution:
${\frac {7^{5}}{49^{3}}}={\frac {7^{5}}{(7^{2})^{3}}}={\frac {7^{5}}{7^{2.3}}}={\frac {7^{5}}{7^{6}}}=7^{5-6}=7^{-1}={\frac {1}{7}}$ c) $({\frac {1}{4}})^{2}(4^{2})$ Solution:
$({\frac {1}{4}})^{2}(4^{2})=(4^{-2})(4^{2})=4^{-2+2}=4^{0}=1$ ## Graphs of Exponential Functions

The graph of the exponential function $a^{x}$ where $a>0,a\neq 1$ always goes through the point $(0,1)$ and has a horizontal asymptote $y=0$ 