To solve the equation [tex]\(\log _3\left(\frac{1}{81}\right)=y\)[/tex], we need to determine the value of [tex]\(y\)[/tex] that satisfies the logarithmic equation. Here are the detailed steps:
1. Rewrite the logarithmic equation in its exponential form:
[tex]\[
\log _3\left(\frac{1}{81}\right) = y \quad \text{implies} \quad 3^y = \frac{1}{81}
\][/tex]
2. Recognize that [tex]\(81\)[/tex] is a power of [tex]\(3\)[/tex]. Specifically:
[tex]\[
81 = 3^4
\][/tex]
3. Substitute this expression into the equation:
[tex]\[
3^y = \frac{1}{3^4}
\][/tex]
4. Recall that [tex]\(\frac{1}{3^4}\)[/tex] can be written as [tex]\(3^{-4}\)[/tex]. Thus:
[tex]\[
3^y = 3^{-4}
\][/tex]
5. Since the bases are the same, we can equate the exponents:
[tex]\[
y = -4
\][/tex]
Therefore, the value of [tex]\(y\)[/tex] that satisfies [tex]\(\log _3\left(\frac{1}{81}\right)=y\)[/tex] is [tex]\(\boxed{-4}\)[/tex].
