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