Precalculus Examples

$\ln (x^{2} - 4 y^{2}) - 2 \ln (x - 2 y) - \ln (7)$

Step 1

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

$\ln (\frac{x^{2} - 4 y^{2}}{7}) - 2 \ln (x - 2 y)$

Step 2

Simplify each term.

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Step 2.1

Simplify the numerator.

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Step 2.1.1

Rewrite $4 y^{2}$ as ${(2 y)}^{2}$ .

$\ln (\frac{x^{2} - {(2 y)}^{2}}{7}) - 2 \ln (x - 2 y)$

Step 2.1.2

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 $b = 2 y$ .

$\ln (\frac{(x + 2 y) (x - (2 y))}{7}) - 2 \ln (x - 2 y)$

Step 2.1.3

Multiply $2$ by $- 1$ .

$\ln (\frac{(x + 2 y) (x - 2 y)}{7}) - 2 \ln (x - 2 y)$

Step 2.2

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

$\ln (\frac{(x + 2 y) (x - 2 y)}{7}) - \ln ({(x - 2 y)}^{2})$

Step 3

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

$\ln (\frac{\frac{(x + 2 y) (x - 2 y)}{7}}{{(x - 2 y)}^{2}})$

Step 4

Multiply the numerator by the reciprocal of the denominator.

$\ln (\frac{(x + 2 y) (x - 2 y)}{7} \cdot \frac{1}{{(x - 2 y)}^{2}})$

Step 5

Combine.

$\ln (\frac{(x + 2 y) (x - 2 y) \cdot 1}{7 {(x - 2 y)}^{2}})$

Step 6

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

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Step 6.1

Factor $x - 2 y$ out of $(x + 2 y) (x - 2 y) \cdot 1$ .

$\ln (\frac{(x - 2 y) ((x + 2 y) \cdot 1)}{7 {(x - 2 y)}^{2}})$

Step 6.2

Cancel the common factors.

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Step 6.2.1

Factor $x - 2 y$ out of $7 {(x - 2 y)}^{2}$ .

$\ln (\frac{(x - 2 y) ((x + 2 y) \cdot 1)}{(x - 2 y) (7 (x - 2 y))})$

Step 6.2.2

Cancel the common factor.

$\ln (\frac{(x - 2 y) ((x + 2 y) \cdot 1)}{(x - 2 y) (7 (x - 2 y))})$

Step 6.2.3

Rewrite the expression.

$\ln (\frac{(x + 2 y) \cdot 1}{7 (x - 2 y)})$

Step 7

Multiply $x + 2 y$ by $1$ .

$\ln (\frac{x + 2 y}{7 (x - 2 y)})$