Question

evaluate the expression without using a calculator.
\log_{6} \frac{1}{\sqrt{6}}
\bigcirc -\frac{1}{2}
\bigcirc -\frac{1}{6}
\bigcirc \frac{1}{6}
\bigcirc \frac{1}{2}

Explanation:

Step1: Rewrite the argument

Rewrite \(\frac{1}{\sqrt{6}}\) as \(6^{-\frac{1}{2}}\) since \(\sqrt{6}=6^{\frac{1}{2}}\) and \(\frac{1}{a^b}=a^{-b}\). So the expression becomes \(\log_{6}6^{-\frac{1}{2}}\).

Step2: Apply log property

Use the logarithm property \(\log_{a}a^{x}=x\) (where \(a>0,a
eq1\)). Here \(a = 6\) and \(x=-\frac{1}{2}\), so \(\log_{6}6^{-\frac{1}{2}}=-\frac{1}{2}\).

Answer:

\(-\frac{1}{2}\) (corresponding to the option \(-\frac{1}{2}\))