Precalculus Examples

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

Step 1

Simplify the numerator.

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Step 1.1

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = \log (x)$ and $b = \log (y)$ .

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

Step 1.2

Simplify.

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Step 1.2.1

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

$\frac{\log (x y) (\log (x) - \log (y))}{\log (x y)}$

Step 1.2.2

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

$\frac{\log (x y) \log (\frac{x}{y})}{\log (x y)}$

Step 2

Cancel the common factor of $\log (x y)$ .

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Step 2.1

Cancel the common factor.

$\frac{\log (x y) \log (\frac{x}{y})}{\log (x y)}$

Step 2.2

Divide $\log (\frac{x}{y})$ by $1$ .

$\log (\frac{x}{y})$