$$ 1 \leftarrow \begin{array}{l} ext { Watch the video and then solve the problem given below. } \ ext { Click here to watch the video. } \ ext { Evaluate the following expression without using a calculator. } \ \log _{6}(\sqrt{6}) \ \log _{6}(\sqrt{6})=\square\end{array} $$

Question

$$ 1 \leftarrow \begin{array}{l}	ext { Watch the video and then solve the problem given below. } \ 	ext { Click here to watch the video. } \ 	ext { Evaluate the following expression without using a calculator. } \ \log _{6}(\sqrt{6}) \ \log _{6}(\sqrt{6})=\square\end{array} $$

UpStudy AI · Accepted Answer

We start with the expression:

$$
\log_{6}(\sqrt{6})
$$

Recall that the square root $$ \sqrt{6} $$ can be written as an exponent:

$$
\sqrt{6} = 6^{\frac{1}{2}}
$$

Substitute this into the logarithm:

$$
\log_{6}(6^{\frac{1}{2}})
$$

Using the property of logarithms $$\log_b (b^x) = x$$, we have:

$$
\log_{6}(6^{\frac{1}{2}}) = \frac{1}{2}
$$

Thus, the final answer is:

$$
\boxed{\frac{1}{2}}
$$
$$ \log_{6}(\sqrt{6}) = \frac{1}{2} $$

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