To solve the expression [tex]\(\log_6\left(\frac{1}{216}\right)\)[/tex], let's break it down step by step:
1. Understand the expression: We are given [tex]\(\log_6\left(\frac{1}{216}\right)\)[/tex], which asks for the exponent to which we must raise the base 6 to obtain [tex]\(\frac{1}{216}\)[/tex].
2. Rewrite the logarithm using properties of exponents:
[tex]\[
\log_6\left(\frac{1}{216}\right) = \log_6 \left(216^{-1}\right)
\][/tex]
This uses the property [tex]\( \frac{1}{a} = a^{-1} \)[/tex].
3. Apply the logarithm power rule: The power rule for logarithms states [tex]\(\log_b(a^c) = c \cdot \log_b(a)\)[/tex].
[tex]\[
\log_6 \left(216^{-1}\right) = -1 \cdot \log_6(216)
\][/tex]
So, the expression simplifies to:
[tex]\[
\log_6 \left(\frac{1}{216}\right) = -\log_6(216)
\][/tex]
4. Evaluate [tex]\(\log_6(216)\)[/tex]:
We need to think about what power 6 must be raised to in order to equal 216. We check:
[tex]\[
6^1 = 6
\][/tex]
[tex]\[
6^2 = 36
\][/tex]
[tex]\[
6^3 = 216
\][/tex]
Since [tex]\(6^3 = 216\)[/tex], we can write:
[tex]\[
\log_6(216) = 3
\][/tex]
5. Combine the results:
[tex]\[
\log_6 \left(\frac{1}{216}\right) = -\log_6(216) = -3
\][/tex]
So, the answer is:
[tex]\[
\boxed{-3}
\][/tex]
