Simplify/Condense log base 3 of 4+2 log base 3 of 1/2+ log base 3 of x

$\log_{3} (4) + 2 \log_{3} (\frac{1}{2}) + \log_{3} (x)$

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

$2 \log_{3} (\frac{1}{2}) + \log_{3} (4 x)$

Simplify each term.

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

$\log_{3} ({(\frac{1}{2})}^{2}) + \log_{3} (4 x)$

Apply the product rule to $\frac{1}{2}$ .

$\log_{3} (\frac{1^{2}}{2^{2}}) + \log_{3} (4 x)$

One to any power is one.

$\log_{3} (\frac{1}{2^{2}}) + \log_{3} (4 x)$

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

$\log_{3} (\frac{1}{4}) + \log_{3} (4 x)$

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

$\log_{3} (\frac{1}{4} (4 x))$

Cancel the common factor of $4$ .

Factor $4$ out of $4 x$ .

$\log_{3} (\frac{1}{4} (4 (x)))$

Cancel the common factor.

$\log_{3} (\frac{1}{4} (4 x))$

Rewrite the expression.

$\log_{3} (x)$