Let's solve the equation [tex]\(3^{2x} = 9^{3x - 4}\)[/tex] step-by-step.
First, we know that [tex]\(9\)[/tex] can be written as a power of [tex]\(3\)[/tex]:
[tex]\[ 9 = 3^2 \][/tex]
Substituting [tex]\(3^2\)[/tex] for [tex]\(9\)[/tex] in the original equation, we get:
[tex]\[ 3^{2x} = (3^2)^{3x - 4} \][/tex]
Next, we use the power of a power rule [tex]\((a^m)^n = a^{m \cdot n}\)[/tex]:
[tex]\[ 3^{2x} = 3^{2 \cdot (3x - 4)} \][/tex]
Simplify the exponent on the right-hand side:
[tex]\[ 3^{2x} = 3^{6x - 8} \][/tex]
Since the bases are the same, we can set the exponents equal to each other:
[tex]\[ 2x = 6x - 8 \][/tex]
To solve for [tex]\(x\)[/tex], we need to isolate [tex]\(x\)[/tex] on one side of the equation. First, subtract [tex]\(6x\)[/tex] from both sides:
[tex]\[ 2x - 6x = -8 \][/tex]
[tex]\[ -4x = -8 \][/tex]
Next, divide both sides by [tex]\(-4\)[/tex]:
[tex]\[ x = \frac{-8}{-4} \][/tex]
[tex]\[ x = 2 \][/tex]
Therefore, the value of [tex]\(x\)[/tex] that satisfies the equation [tex]\(3^{2x} = 9^{3x - 4}\)[/tex] is [tex]\(x = 2\)[/tex].
So, the answer is:
[tex]\[ \boxed{2} \][/tex]
