9^x=3^(x^2-4x)

Question

Solve the exponential equation exactly for x.

Answer 1

First, notice that 9 = 3^2. Then the term on the left can be rewritten as (3²)^x. Using the rules of powers, that becomes 3^2x.

The term on the right is already a power of 3 -- there, the power is x² - 4x.

Since both sides are expressed as powers of 3, you can ignore the 3 and set the exponents equal to each other.

We have:           2x  =  x² - 4x

We can rearrange this into a quadratic equation:     x² - 6x  = 0, or

                                                                               x(x - 6 ) = 0

Setting each factor equal to 0, we get:

     x = 0   or  x  =  6

 

Answer 2

Jessica,

For these kind of problems you want to put the numbers raised to an exponential power to be the same, i.e. if 2^x=8^5, then I want to change 8 to 2^3.  This would make the equation 2^x=(2^3)^5.  Under the rules for exponents, the left hand side becomes 2^15 (3x5), making the equation 2^x=2^15, so x=15 for the left hand side to equal the right hand side.

For your problem, what is the common base between 3 and 9?

Then once you have a common base, you can eliminate the base, and you will have a quadratic equation, which can be set equal to zero and use the zero product property to solve.

I hope this helps.  John