Simplify/Condense log base 6 of 9+2 log base 6 of 1/3-3 log base 6 of x

$\log_{6} (9) + 2 \log_{6} (\frac{1}{3}) - 3 \log_{6} (x)$

Simplify each term.

Simplify $2 \log_{6} (\frac{1}{3})$ by moving $2$ inside the logarithm.

$\log_{6} (9) + \log_{6} ({(\frac{1}{3})}^{2}) - 3 \log_{6} (x)$

Apply the product rule to $\frac{1}{3}$ .

$\log_{6} (9) + \log_{6} (\frac{1^{2}}{3^{2}}) - 3 \log_{6} (x)$

One to any power is one.

$\log_{6} (9) + \log_{6} (\frac{1}{3^{2}}) - 3 \log_{6} (x)$

Raise $3$ to the power of $2$ .

$\log_{6} (9) + \log_{6} (\frac{1}{9}) - 3 \log_{6} (x)$

Simplify $- 3 \log_{6} (x)$ by moving $3$ inside the logarithm.

$\log_{6} (9) + \log_{6} (\frac{1}{9}) - \log_{6} (x^{3})$

Use the product property of logarithms, $\log_{b} (x) + \log_{b} (y) = \log_{b} (x y)$ .

$\log_{6} (9 (\frac{1}{9})) - \log_{6} (x^{3})$

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$\log_{6} (\frac{9 (\frac{1}{9})}{x^{3}})$

Combine $9$ and $\frac{1}{9}$ .

$\log_{6} (\frac{\frac{9}{9}}{x^{3}})$

Divide $9$ by $9$ .

$\log_{6} (\frac{1}{x^{3}})$