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 $y=0$

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