Sure, let's solve the equation [tex]\(3^{2x} = \frac{1}{81}\)[/tex] step-by-step.
1. Understand the components of the equation:
- We have an exponential equation where the base is [tex]\(3\)[/tex].
2. Rewrite [tex]\(\frac{1}{81}\)[/tex] in terms of base 3:
- Notice that [tex]\(81 = 3^4\)[/tex]. Hence, [tex]\(\frac{1}{81} = \frac{1}{3^4} = 3^{-4}\)[/tex].
Now the equation becomes:
[tex]\[
3^{2x} = 3^{-4}
\][/tex]
3. Set the exponents equal to each other:
- Since the bases are the same, we can equate the exponents:
[tex]\[
2x = -4
\][/tex]
4. Solve for [tex]\(x\)[/tex]:
- Divide both sides of the equation by 2 to isolate [tex]\(x\)[/tex]:
[tex]\[
x = \frac{-4}{2}
\][/tex]
[tex]\[
x = -2
\][/tex]
So, the solution to the equation [tex]\(3^{2x} = \frac{1}{81}\)[/tex] is:
[tex]\[
x = -2
\][/tex]
