5.1 Exponential Functions, 6.6 Logarithmic and Exponential Equations, Exponential Equations

Solving an Exponential Equation: Relating the Bases

Example:  Solve the exponential equation.

4^{x-9}=1/1024

Solution:

The exponential equation
4^{x-9}=1/1024
Try to write both sides of the equation with the same base.  Try 4 since there is a base of 4 on the left 4^{x-9}=1/4^5
Using a property of negative exponents move the base to the numerator  4^{x-9}=4^{-5}
Now that that the bases are the same the exponents must be equal  4^{x-9}=4^{-5}

x-9=-5
Solve for x
  x-9=-5

x-9+9=-5+9

x=4

The solution the the exponential equation is 4.

Here is a youtube video with a similar example.