Precalculus Examples

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

Step 1

Simplify each term.

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Step 1.1

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

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

Step 1.2

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

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

Step 1.3

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

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Step 1.3.1

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

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

Step 1.3.2

Apply the product rule to $2 x$ .

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

Step 1.4

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

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

Step 1.5

Simplify each term.

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Step 1.5.1

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

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

Step 1.5.2

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

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

Step 1.6

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

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

Step 1.7

Multiply $3$ by $25$ .

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

Step 2

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

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

Step 3

Multiply the numerator by the reciprocal of the denominator.

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

Step 4

Combine.

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

Step 5

Simplify the expression.

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Step 5.1

Multiply $4$ by $1$ .

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

Step 5.2

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

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