To solve the exponential equation [tex]\(4^{x-2} = \frac{1}{4}\)[/tex], follow these steps:
1. Rewrite [tex]\(\frac{1}{4}\)[/tex] to a common base: Recall that [tex]\(\frac{1}{4}\)[/tex] can be written as an exponential expression with base 4. Specifically, [tex]\(\frac{1}{4}\)[/tex] is the same as [tex]\(4^{-1}\)[/tex].
[tex]\[
4^{x-2} = 4^{-1}
\][/tex]
2. Equalize the exponents: Since the bases on both sides of the equation are the same, we can set the exponents equal to each other.
[tex]\[
x - 2 = -1
\][/tex]
3. Solve for [tex]\(x\)[/tex]: Now, solve the equation for [tex]\(x\)[/tex].
[tex]\[
x - 2 = -1
\][/tex]
Add 2 to both sides of the equation to isolate [tex]\(x\)[/tex]:
[tex]\[
x = -1 + 2
\][/tex]
Simplify the right-hand side:
[tex]\[
x = 1
\][/tex]
Therefore, the solution to the equation [tex]\(4^{x-2} = \frac{1}{4}\)[/tex] is [tex]\(x = 1\)[/tex].
Thus, the correct answer is:
B) [tex]\(x=1\)[/tex]
