# Solving an Exponential Equation: Relating the Bases

Example:  Solve the exponential equation.

Solution:

 The exponential equation Try to write both sides of the equation with the same base.  Try 4 since there is a base of 4 on the left Using a property of negative exponents move the base to the numerator Now that that the bases are the same the exponents must be equal Solve for x

The solution the the exponential equation is 4.

# Graphing an Exponential Equation by Transformations

Example: For the function below.  Graph using transformations.  Find the y-intercept.  State the horizontal asymptote and the domain and range.

First we must examine the base function

Graph using plotting points.  We can use the standard set of x-values to find ordered pairs.

xy
-22^(-2)=1/4
-12^(-1)=1/2
02^0=1
12^1=2
22^2=4

The graph below shows the points plotted and the line that connects them.  This graph has a horizontal asymptote at y=0.  The domain is   and the range is

Analyze the transformations.

The +2 in the exponent shifts the graph left 2 units.

The – in the front of the base reflects the graph over the x-axis.

The +2 next to the base shifts the graph and the horizontal asymptote up two units.

You can see the graph after the transformations.

The horizontal asymptote is y=2.  The domain is   and the range is

To find the y-intercept we let x=0.

Thus the y-intercept is (0,-2)