Answer:
\(\text{log}_{\frac{1}{6}}(36)=-2\)
Step-by-step explanation:
We have been given a logarithm expression \(\text{log}_{\frac{1}{6}}(36)\). We are asked to evaluate our given expression.
Using logarithm rule \(log_{\frac{1}{a}}(x)=-log_a(x)\), we can write our expression as:
\(\text{log}_{\frac{1}{6}}(36)=-\text{log}_6(36)\)
\(\text{log}_{\frac{1}{6}}(36)=-\text{log}_6(6^2)\)
Using logarithm rule \(\text{log}_{a}(x^b)=b\text{log}_a(x)\), we will get:
\(\text{log}_{\frac{1}{6}}(36)=-2\text{log}_6(6)\)
Applying rule \(\text{log}_{a}(a)=1\), we will get:
\(\text{log}_{\frac{1}{6}}(36)=-2\cdot 1\)
\(\text{log}_{\frac{1}{6}}(36)=-2\)
Therefore, \(\text{log}_{\frac{1}{6}}(36)=-2\).
