Simplify/Condense 1/2 log base 5 of 16-3 log base 5 of x+4 log base 5 of y

$\frac{1}{2} \log_{5} (16) - 3 \log_{5} (x) + 4 \log_{5} (y)$

Simplify each term.

Simplify $\frac{1}{2} \log_{5} (16)$ by moving $\frac{1}{2}$ inside the logarithm.

$\log_{5} (16^{\frac{1}{2}}) - 3 \log_{5} (x) + 4 \log_{5} (y)$

Rewrite $16$ as $4^{2}$ .

$\log_{5} ({(4^{2})}^{\frac{1}{2}}) - 3 \log_{5} (x) + 4 \log_{5} (y)$

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

$\log_{5} (4^{2 (\frac{1}{2})}) - 3 \log_{5} (x) + 4 \log_{5} (y)$

Cancel the common factor of $2$ .

Cancel the common factor.

$\log_{5} (4^{2 (\frac{1}{2})}) - 3 \log_{5} (x) + 4 \log_{5} (y)$

Rewrite the expression.

$\log_{5} (4^{1}) - 3 \log_{5} (x) + 4 \log_{5} (y)$

Evaluate the exponent.

$\log_{5} (4) - 3 \log_{5} (x) + 4 \log_{5} (y)$

Simplify $- 3 \log_{5} (x)$ by moving $3$ inside the logarithm.

$\log_{5} (4) - \log_{5} (x^{3}) + 4 \log_{5} (y)$

Simplify $4 \log_{5} (y)$ by moving $4$ inside the logarithm.

$\log_{5} (4) - \log_{5} (x^{3}) + \log_{5} (y^{4})$

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

$\log_{5} (\frac{4}{x^{3}}) + \log_{5} (y^{4})$

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

$\log_{5} (\frac{4}{x^{3}} y^{4})$

Combine $\frac{4}{x^{3}}$ and $y^{4}$ .

$\log_{5} (\frac{4 y^{4}}{x^{3}})$