To solve the equation [tex]\(3^{2x} = \frac{1}{3}\)[/tex] for [tex]\(x\)[/tex], follow these detailed steps:
1. Understand the equation:
The equation given is [tex]\(3^{2x} = \frac{1}{3}\)[/tex].
2. Express [tex]\(\frac{1}{3}\)[/tex] as a power of 3:
[tex]\(\frac{1}{3}\)[/tex] can be written as [tex]\(3^{-1}\)[/tex] because [tex]\(3^{-1} = \frac{1}{3}\)[/tex].
So, the equation becomes:
[tex]\[
3^{2x} = 3^{-1}
\][/tex]
3. Set the exponents equal to each other:
Since the bases are the same (both are 3), we can set the exponents equal to each other:
[tex]\[
2x = -1
\][/tex]
4. Solve for [tex]\(x\)[/tex]:
Divide both sides of the equation by 2 to isolate [tex]\(x\)[/tex]:
[tex]\[
x = \frac{-1}{2} = -\frac{1}{2}
\][/tex]
Thus, the solution to the equation [tex]\(3^{2x} = \frac{1}{3}\)[/tex] is [tex]\(x = -\frac{1}{2}\)[/tex].
Therefore, the correct answer is:
D. [tex]\(-\frac{1}{2}\)[/tex]
