Precalculus Examples

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

Step 1

Simplify $\frac{3}{2} \log_{2} (x + y)$ by moving $\frac{3}{2}$ inside the logarithm.

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

Step 2

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

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

Step 3

Simplify each term.

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Step 3.1

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

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

Step 3.2

Move ${(x + y)}^{1}$ to the numerator using the negative exponent rule $\frac{1}{b^{n}} = b^{- n}$ .

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

Step 3.3

Multiply ${(x + y)}^{\frac{3}{2}}$ by ${(x + y)}^{- 1}$ by adding the exponents.

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Step 3.3.1

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 3.3.2

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 3.3.3

Combine $- 1$ and $\frac{2}{2}$ .

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

Step 3.3.4

Combine the numerators over the common denominator.

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

Step 3.3.5

Simplify the numerator.

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Step 3.3.5.1

Multiply $- 1$ by $2$ .

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

Step 3.3.5.2

Subtract $2$ from $3$ .

$\log_{2} (\frac{{(x + y)}^{\frac{1}{2}}}{x - y}) + \log (x - y)$