# Math 22 Exponential Functions

## Definition of Exponential Function

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


## Properties of Exponents

Let ${\displaystyle a}$ and ${\displaystyle b}$ be positive real numbers, and let ${\displaystyle x}$ and ${\displaystyle y}$ be real numbers.

1.${\displaystyle a^{0}=1}$

2.${\displaystyle a^{x}a^{y}=a^{x+y}}$

3.${\displaystyle {\frac {a^{x}}{a^{y}}}=a^{x-y}}$

4.${\displaystyle (a^{x})^{y}=a^{xy}}$

5.${\displaystyle (ab)^{x}=a^{x}b^{x}}$

6.${\displaystyle ({\frac {a}{b}})^{x}={\frac {a^{x}}{b^{x}}}}$

7.${\displaystyle a^{-x}={\frac {1}{a^{x}}}}$

Exercises Use the properties of exponents to simplify each expression:

a) ${\displaystyle (8^{\frac {1}{2}})(2^{\frac {1}{2}})}$

Solution:
${\displaystyle (8^{\frac {1}{2}})(2^{\frac {1}{2}})=(8\cdot 2)^{\frac {1}{2}}=16^{\frac {1}{2}}}$

b) ${\displaystyle {\frac {7^{5}}{49^{3}}}}$

Solution:
${\displaystyle {\frac {7^{5}}{49^{3}}}={\frac {7^{5}}{(7^{2})^{3}}}={\frac {7^{5}}{7^{2.3}}}={\frac {7^{5}}{7^{6}}}=7^{5-6}=7^{-1}={\frac {1}{7}}}$

c) ${\displaystyle ({\frac {1}{4}})^{2}(4^{2})}$

Solution:
${\displaystyle ({\frac {1}{4}})^{2}(4^{2})=(4^{-2})(4^{2})=4^{-2+2}=4^{0}=1}$

## Graphs of Exponential Functions

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