Expand the Logarithmic Expression log base 6 of square root of 3mn^8

$\log_{6} (\sqrt{3 m n^{8}})$

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3 m n^{8}}$ as ${(3 m n^{8})}^{\frac{1}{2}}$ .

$\log_{6} ({(3 m n^{8})}^{\frac{1}{2}})$

Expand $\log_{6} ({(3 m n^{8})}^{\frac{1}{2}})$ by moving $\frac{1}{2}$ outside the logarithm.

$\frac{1}{2} \log_{6} (3 m n^{8})$

Rewrite $\log_{6} (3 m n^{8})$ as $\log_{6} (3 m) + \log_{6} (n^{8})$ .

$\frac{1}{2} (\log_{6} (3 m) + \log_{6} (n^{8}))$

Rewrite $\log_{6} (3 m)$ as $\log_{6} (3) + \log_{6} (m)$ .

$\frac{1}{2} (\log_{6} (3) + \log_{6} (m) + \log_{6} (n^{8}))$

Expand $\log_{6} (n^{8})$ by moving $8$ outside the logarithm.

$\frac{1}{2} (\log_{6} (3) + \log_{6} (m) + 8 \log_{6} (n))$

Apply the distributive property.

$\frac{1}{2} \log_{6} (3) + \frac{1}{2} \log_{6} (m) + \frac{1}{2} (8 \log_{6} (n))$

Simplify.

Combine $\frac{1}{2}$ and $\log_{6} (3)$ .

$\frac{\log_{6} (3)}{2} + \frac{1}{2} \log_{6} (m) + \frac{1}{2} (8 \log_{6} (n))$

Combine $\frac{1}{2}$ and $\log_{6} (m)$ .

$\frac{\log_{6} (3)}{2} + \frac{\log_{6} (m)}{2} + \frac{1}{2} (8 \log_{6} (n))$

Cancel the common factor of $2$ .

Factor $2$ out of $8 \log_{6} (n)$ .

$\frac{\log_{6} (3)}{2} + \frac{\log_{6} (m)}{2} + \frac{1}{2} (2 (4 \log_{6} (n)))$

Cancel the common factor.

$\frac{\log_{6} (3)}{2} + \frac{\log_{6} (m)}{2} + \frac{1}{2} (2 (4 \log_{6} (n)))$

Rewrite the expression.

$\frac{\log_{6} (3)}{2} + \frac{\log_{6} (m)}{2} + 4 \log_{6} (n)$