To solve the equation [tex]\(4^x = 8^{x-1}\)[/tex], we need to find the value of [tex]\(x\)[/tex] that satisfies this equation. Let's go through the steps to solve it:
1. Express Both Sides Using the Same Base:
We know that [tex]\(4\)[/tex] can be written as [tex]\(2^2\)[/tex] and [tex]\(8\)[/tex] can be written as [tex]\(2^3\)[/tex]. Therefore, we can rewrite the equation:
[tex]\[
(2^2)^x = (2^3)^{x-1}
\][/tex]
2. Simplify the Exponents:
Using the power rule [tex]\((a^m)^n = a^{m \cdot n}\)[/tex], we can simplify both sides:
[tex]\[
2^{2x} = 2^{3(x-1)}
\][/tex]
3. Set the Exponents Equal to Each Other:
Since the bases are the same, we can set the exponents equal to each other:
[tex]\[
2x = 3(x - 1)
\][/tex]
4. Solve the Resulting Linear Equation:
Distribute and simplify the equation:
[tex]\[
2x = 3x - 3
\][/tex]
Subtract [tex]\(3x\)[/tex] from both sides to isolate [tex]\(x\)[/tex]:
[tex]\[
-x = -3
\][/tex]
Divide both sides by [tex]\(-1\)[/tex]:
[tex]\[
x = 3
\][/tex]
5. Check the Possible Answers:
The provided options were:
- [tex]\(x = \frac{3}{4}\)[/tex]
- [tex]\(x = 1\)[/tex]
- [tex]\(x = 3\)[/tex]
- [tex]\(x = 6\)[/tex]
We found that [tex]\(x = 3\)[/tex] satisfies the equation [tex]\(4^x = 8^{x-1}\)[/tex].
Therefore, the correct answer is:
[tex]\[
\boxed{3}
\][/tex]
