As we’ve learned, there are a multitude of situations that can be modeled by exponential functions, such as investment growth, radioactive decay, atmospheric pressure changes, and temperatures of a cooling object.

By definition, an exponential function has a constant as a base and an independent variable as an exponent. Thus, g(x)=x3 does not represent an exponential function because the base is an independent variable.

Therefore, as our practical case of exponential functions shows, an exponential function cannot have a base of 0, 1, or a negative value.

Exponential equations are equations in which variables occur as exponents. For example, exponential equations are in the form ax=by . In other words, if the bases are the same, then the exponents must be equal. Example 1: Solve the equation 42x−1=64 .

How To: Given an exponential function of the form f(x)=bx f ( x ) = b x , graph the function

1. Create a table of points.
2. Plot at least 3 point from the table including the y-intercept (0,1) .
3. Draw a smooth curve through the points.
4. State the domain, (−∞,∞) , the range, (0,∞) , and the horizontal asymptote, y=0 .

Exponential regression is used to model situations in which growth begins slowly and then accelerates rapidly without bound, or where decay begins rapidly and then slows down to get closer and closer to zero. We use the command “ExpReg” on a graphing utility to fit an exponential function to a set of data points.

In linear regression, the function is a linear (straight-line) equation. In power or exponential regression, the function is a power (polynomial) equation of the form or an exponential function in the form .