To solve the equation [tex]\(3^{4x} = 27^{x-3}\)[/tex], we can follow these steps:
1. Express 27 with base 3:
We know that [tex]\(27\)[/tex] can be written as [tex]\(3^3\)[/tex]. Thus, we can rewrite the equation using a common base:
[tex]\[
3^{4x} = (3^3)^{x-3}
\][/tex]
2. Simplify the right side:
Apply the power of a power property [tex]\(((a^m)^n = a^{m \cdot n})\)[/tex]:
[tex]\[
3^{4x} = 3^{3(x-3)}
\][/tex]
3. Equate the exponents:
Since the bases are the same, we can set the exponents equal to each other:
[tex]\[
4x = 3(x-3)
\][/tex]
4. Solve for [tex]\(x\)[/tex]:
Distribute the [tex]\(3\)[/tex] on the right side:
[tex]\[
4x = 3x - 9
\][/tex]
Next, isolate [tex]\(x\)[/tex] by subtracting [tex]\(3x\)[/tex] from both sides:
[tex]\[
4x - 3x = -9
\][/tex]
Simplify the left side:
[tex]\[
x = -9
\][/tex]
So, the value of [tex]\(x\)[/tex] that satisfies [tex]\(3^{4x} = 27^{x-3}\)[/tex] is [tex]\(\boxed{-9}\)[/tex].
