To solve the equation [tex]\(3^{4x} = 27^{x-3}\)[/tex], we need to manipulate and simplify the expressions. Here's a step-by-step solution:
1. Rewrite [tex]\(27\)[/tex] as a power of [tex]\(3\)[/tex]: We know that [tex]\(27 = 3^3\)[/tex]. Thus, the equation can be rewritten as:
[tex]\[
3^{4x} = (3^3)^{x-3}
\][/tex]
2. Simplify the right-hand side: When raising a power to another power, you multiply the exponents. So, we get:
[tex]\[
3^{4x} = 3^{3(x-3)}
\][/tex]
3. Simplify the exponent on the right-hand side: Multiply inside the parentheses:
[tex]\[
3^{4x} = 3^{3x - 9}
\][/tex]
4. Set the exponents equal to each other: Since the bases are the same (both are base [tex]\(3\)[/tex]), we can set the exponents equal to each other:
[tex]\[
4x = 3x - 9
\][/tex]
5. Solve for [tex]\(x\)[/tex]: Subtract [tex]\(3x\)[/tex] from both sides of the equation to isolate [tex]\(x\)[/tex]:
[tex]\[
4x - 3x = -9
\][/tex]
[tex]\[
x = -9
\][/tex]
Thus, the value of [tex]\(x\)[/tex] that satisfies the equation [tex]\(3^{4x} = 27^{x-3}\)[/tex] is [tex]\(-9\)[/tex].
Therefore, the correct answer is [tex]\(\boxed{-9}\)[/tex].
