Solve for x in the equation 3^(2x - 1) = 3 * 9^(2x + 6). What is the value of x?

To solve the equation ( 3^{2x - 1} = 3 \cdot 9^{2x + 6} ), we start by rewriting the term involving ( 9 ). Since ( 9 ) can be expressed as ( 3^2 ), we can rewrite ( 9^{2x + 6} ) as ( (3^2)^{2x + 6} ) or ( 3^{2(2x + 6)} ). This simplifies to ( 3^{4x + 12} ).

Now, the equation looks like this:

[ 3^{2x - 1} = 3 \cdot 3^{4x + 12} ]

We can also simplify the right side by combining the exponents of ( 3 ):

[ 3^{2x - 1} = 3^{1 + 4x + 12} ]

[ 3^{2x - 1} = 3^{4x + 13} ]

Since the bases are the same, we can set the exponents equal to each other:

[ 2x - 1 = 4