 # Exponential Functions #### Level 1 to Level 4

##### Exponent Laws

A power consists of a base and an exponent. For the power, $x^2$, the base is $x$ and the exponent is $2$. There are eight “Exponent Laws” that can be used to simplify larger and complex exponential expressions. These are the first seven.

#### Level 5 to Level 6

##### Rational Exponents

If your exponent is a fraction (rational number), then the denominator of that number represents the index of a radical. For example, $x^{2/3}$ can be written as $\sqrt{x^2}$. This is the eighth “Exponent Law”.

#### Level 5 to Level 7

##### Graphing Exponential Functions

The basic exponential function takes the form: $y=b^x$ in which $b$ the base is some number larger than 0 (i.e. $b>0$). There are several distinct properties of exponential functions such as a horizontal asymptote, a line that the function gets close to, but they never touch.

#### Level 5 to Level 7

##### Solving Exponential Equations – Same Base Method

There are two methods to solving exponential functions. The first method involves getting setting two powers equal to each other, then manipulating the two, so that both have the same base. #### Level  7

##### Solving Exponential Equations – Logarithm Method

There are two methods to solving exponential functions. The second method involves setting one power equal to a number then, changing the equation into logarithmic form.

#### Level 5 to Level 6

##### Applications – Exponential Growth and Decay

The half-life of radioactive materials, the increase in population of a small city, the growth of an investment or the depreciation of a vehicle are all examples of an exponential function in nature.