Precalculus Examples

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

Step 1

Simplify each term.

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Step 1.1

Simplify $3 \log (x + y)$ by moving $3$ inside the logarithm.

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

Step 1.2

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

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

Step 2

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

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

Step 3

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

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

Step 4

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 $b = y$ .

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

Step 5

Simplify terms.

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Step 5.1

Cancel the common factor of ${(x + y)}^{3}$ and $x + y$ .

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Step 5.1.1

Factor $x + y$ out of ${(x + y)}^{3} {(x - y)}^{2}$ .

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

Step 5.1.2

Cancel the common factors.

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Step 5.1.2.1

Cancel the common factor.

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

Step 5.1.2.2

Rewrite the expression.

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

Step 5.2

Cancel the common factor of ${(x - y)}^{2}$ and $x - y$ .

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Step 5.2.1

Factor $x - y$ out of ${(x + y)}^{2} {(x - y)}^{2}$ .

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

Step 5.2.2

Cancel the common factors.

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Step 5.2.2.1

Multiply by $1$ .

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

Step 5.2.2.2

Cancel the common factor.

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

Step 5.2.2.3

Rewrite the expression.

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

Step 5.2.2.4

Divide ${(x + y)}^{2} (x - y)$ by $1$ .

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

Step 5.3

Apply the distributive property.

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

Step 5.4

Reorder.

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Step 5.4.1

Rewrite using the commutative property of multiplication.

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

Step 5.4.2

Reorder factors in $\log ({(x + y)}^{2} x - {(x + y)}^{2} y)$ .

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