Write as a Single Logarithm 2( log of 2x+ log of y)-( log of 3+2 log of 5)

$2 (\log (2 x) + \log (y)) - (\log (3) + 2 \log (5))$

Simplify each term.

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

$2 \log (2 x y) - (\log (3) + 2 \log (5))$

Simplify $2 \log (2 x y)$ by moving $2$ inside the logarithm.

$\log ({(2 x y)}^{2}) - (\log (3) + 2 \log (5))$

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

Apply the product rule to $2 x y$ .

$\log ({(2 x)}^{2} y^{2}) - (\log (3) + 2 \log (5))$

Apply the product rule to $2 x$ .

$\log (2^{2} x^{2} y^{2}) - (\log (3) + 2 \log (5))$

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

$\log (4 x^{2} y^{2}) - (\log (3) + 2 \log (5))$

Simplify each term.

Simplify $2 \log (5)$ by moving $2$ inside the logarithm.

$\log (4 x^{2} y^{2}) - (\log (3) + \log (5^{2}))$

Raise $5$ to the power of $2$ .

$\log (4 x^{2} y^{2}) - (\log (3) + \log (25))$

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

$\log (4 x^{2} y^{2}) - \log (3 \cdot 25)$

Multiply $3$ by $25$ .

$\log (4 x^{2} y^{2}) - \log (75)$

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

$\log (\frac{4 x^{2} y^{2}}{75})$