Simplify/Condense 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 quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

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

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

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

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

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

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

Apply the product rule to $2 x$ .

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

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

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

Simplify each term.

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

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

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

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

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

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

Multiply $3$ by $25$ .

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

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

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

Multiply the numerator by the reciprocal of the denominator.

$\log (\frac{4 x^{2}}{y^{2}} \cdot \frac{1}{75})$

Combine.

$\log (\frac{4 x^{2} \cdot 1}{y^{2} \cdot 75})$

Simplify the expression.

Multiply $4$ by $1$ .

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

Move $75$ to the left of $y^{2}$ .

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