Simplify/Condense 1/2( log base 6 of 45- log base 6 of 5)+ log base 6 of 12

$\frac{1}{2} (\log_{6} (45) - \log_{6} (5)) + \log_{6} (12)$

Simplify each term.

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

$\frac{1}{2} \cdot \log_{6} (\frac{45}{5}) + \log_{6} (12)$

Divide $45$ by $5$ .

$\frac{1}{2} \cdot \log_{6} (9) + \log_{6} (12)$

Simplify $\frac{1}{2} \log_{6} (9)$ by moving $\frac{1}{2}$ inside the logarithm.

$\log_{6} (9^{\frac{1}{2}}) + \log_{6} (12)$

Rewrite $9$ as $3^{2}$ .

$\log_{6} ({(3^{2})}^{\frac{1}{2}}) + \log_{6} (12)$

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\log_{6} (3^{2 (\frac{1}{2})}) + \log_{6} (12)$

Cancel the common factor of $2$ .

Cancel the common factor.

$\log_{6} (3^{2 (\frac{1}{2})}) + \log_{6} (12)$

Rewrite the expression.

$\log_{6} (3^{1}) + \log_{6} (12)$

Evaluate the exponent.

$\log_{6} (3) + \log_{6} (12)$

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

$\log_{6} (3 \cdot 12)$

Multiply $3$ by $12$ .

$\log_{6} (36)$

Logarithm base $6$ of $36$ is $2$ .

$2$